Lie Groups

Karl-Heinz Fieseler

Uppsala 2007

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Contents

1 Introduction 3

2 Differentiable Manifolds 6

3 Lie Groups 11

4 Vector Fields 15

5 The Lie Algebra of a Lie Group 25

6 Homogeneous Spaces 30

7 The Exponential Map 34

8 Subgroups and Subalgebras 38

9 Lie Algebras of Dimension ≤ 3 45

10 The Universal Covering Group 51

11 Structure Theory for Lie Algebras 59

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1 Introduction

The groups one investigates in algebra usually are finite or finitely generated,but for example R endowed with the addition of real numbers is far frombeing finitely generated. Or take

GLn(R) := A ∈ Rn,n; det(A) 6= 0,

the group of invertible n× n-matrices with real entries, the group law beingthe multiplication of matrices.

In order to understand even such groups one considers groups with addi-tional structure compatible with the group action.

Definition 1.1. A topological group is a group G endowed with a Hausdorfftopology such that both the group multiplication (group law)

µ : G×G −→ G, (a, b) 7→ ab,

and the ”inversion”ι : G −→ G, a 7→ a−1

are continuous maps.

Example 1.2. 1. The additive group G := R endowed with its standardtopology.

2. The groupG := GLn(R) ⊂ Rn,n ∼= Rn2

,

endowed with the topology as a(n open) subset of Rn2.

Let us comment on the second example: First of all,

GLn(R) = det−1(R∗) ⊂ Rn,n

is an open set of Rn,n as the inverse image of the open set R∗ := R \ 0 (thepunctured real line) with respect to a continuous map, the determinant

det : Rn,n −→ R, A = (αij) 7→ det(A) =∑π∈Sn

sign(π)n∏i=1

αi,π(i),

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a polynomial in the entries αij of A. That matrix multiplication is continuousis immediate, while the continuity of the inversion A 7→ A−1 follows from thefollowing formula

A−1 = (γij) with γij = (−1)i+jdet(Aji)

det(A),

where Ak` ∈ Rn−1,n−1 denotes the matrix obtained from A ∈ Rn,n by deletingthe k-th row and the `-th column. With other words, the entries γij of A−1

are rational functions in the entries of A (with a non-vanishing denominator).

Indeed continuous functions can be quite pathological, while differentiablefunctions are more easily understood, since locally they can be approximatedby linear functions. So it seems natural to look for a notion of ”differentiablegroups”. We give here a very restricted provisional definition:

Definition 1.3. Assume that G ⊂ Rm is both an open subset of Rm anda group (with a suitable group law). It is called a differentiable group ifthe group multiplication as well as the inversion are differentiable (i.e. C∞-)maps.

Example 1.4. 1. The vector space Rm endowed with the addition of vec-tors as group law.

2. GLn(R) with m = n2.

3. The direct product G := Rn ×GLn(R) with m = n+ n2.

4. There is an other way to endow Rn × GLn(R) with the structure of aLie group: Identify a pair (a,B) ∈ Rn×GLn(R) with the ”affine linearmap” Rn 3 x 7→ a + Bx ∈ Rn. The composition of two affine linearmaps being again affine linear, we obtain the following new group law

(a,B)(c,D) := (a+Bc,BD).

on Rn ×GLn(R).

Now the basic idea in the study of differentiable groups is to replace thecommutator map

K : G×G −→ G, (x, y) 7→ xyx−1y−1

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with the ”bilinear part” of its Taylor expansion at (e, e) - here e ∈ G denotesthe neutral element of the group G. Let us explain that: Denote DK(e, e) ∈Rm,2m the Jacobian matrix of K at (e, e) - the linear part of the Taylorexpansion. Then we have for small ξ, η ∈ Rm the expansion

K(e+ ξ, e+ η) = K((e, e) + (ξ, η))

e+DK(e, e)( ξη

)+

∑1≤i,j≤m

∂K

∂xi∂yj(e, e)ξiηj + ....

Now the ”bilinear term”

[ξ, η] :=∑

1≤i,j≤m

∂K

∂xi∂yj(e, e)ξiηj

defines a bilinear map

[.., ..] : Rm × Rm −→ Rm.

It turns out that that map determines the group law near (e, e) ∈ G × Gcompletely, so one can replace the local study of differentiable groups withthe study of certain bilinear maps [.., ..] : Rm × Rm −→ Rm.

Let us discuss the example G = GLn(R) ⊂ Rn,n and compute the map[.., ..] : Rn,n × Rn,n −→ Rn,n. For a matrix A = (αij) we define its norm by

||A|| := nmax|αij|; 1 ≤ i, j ≤ n

and note that it is even well behaved with respect to products: ||AB|| ≤||A|| · ||B||. Now denote E ∈ GLn(R) the unit matrix (replacing e ∈ G)and take X, Y ∈ Rn,n of norm < 1 (replacing ξ and η). Then we haveE +X ∈ GLn(R) with

(E +X)−1 = E −X +X2 −X3 + ...,

a convergent series. Consequently

K(E +X,E + Y ) = (E +X)(E + Y )(E −X +X2 − ....)(E − Y + Y 2 − ...)

= E+(X+Y −X−Y )+(X2 +Y 2 +XY −X2−XY −Y X−Y 2 +XY )+ ...

with the dots representing terms of total degree > 2 in X and Y . Thus thelinear term vanishes and

[X, Y ] = XY − Y X

is the commutator of the matrices X, Y ∈ Rn,n.

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2 Differentiable Manifolds

In Definition 1.3 we require G ⊂ Rm (as an open subset) for a differentiablegroup. That is a very restrictive condition. E.g. it holds for G := C∗ ⊂ C ∼=R2, the multiplicative group of non-zero complex numbers, but it does nothold any longer for its closed subgroup

S1 := z ∈ C; |z| = 1,

the unit circle. (An open subset of Rm is never compact.) Nevertheless itlocally looks like the real line: There are homeomorphisms

ψ1 : V1 :=]− π, π[−→ U1 := S1 \ −1, x 7→ eix

andψ2 : V2 :=]0, 2π[−→ U2 := S1 \ 1, x 7→ eix.

Thus we are led to the notion of a topological manifold:

Definition 2.1. An m-dimensional topological manifold M is a Hausdorfftopological space admitting an open cover

M =⋃i∈I

Ui

with open subset Ui ⊂M homeomorphic to open subsets Vi ⊂ Rm.

Example 2.2. 1. Any open subset of Rm is an m-dimensional topologicalmanifold.

2. M := S1 = U1 ∪ U2 is a one dimensional topological manifold.

3. Denote ||x|| =√x2

1 + ...+ x2n+1 the euclidean norm of a vector x ∈

Rn+1. Then the sphere

Sn := x ∈ Rn+1; ||x|| = 1

is an n-dimensional topological manifold: We have

Sn = U1 ∪ U2

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with U1 := Sn \ −en+1, U2 := Sn \ en+1, where Ui ∼= Rn. Forexample the maps

σi : Ui −→ Rn, x = (x′, xn+1) 7→ x′

1− (−1)ixn+1

, i = 1, 2

are homeomorphisms: For x ∈ Ui the point (σi(x), 0) is the intersectionof the line spanned by x and −en+1 (for i = 1) resp. en+1 (for i = 2)with the hyperplane Rn × 0.

Now one could try to study a function f : M −→ R on a topologicalmanifold M by considering what one gets by composing f with inverse home-

omorphisms Rm ⊃ Vψ−→ U ⊂ M and then apply analysis to the composite

f ψ : V −→ R. But then it will depend on the choice of the homeomorphismψ, whether f ψ is differentiable or not. One can avoid that difficulty byrestricting to a system, ”atlas”, of ”mutually compatible” homeomorphisms,also called ”charts”:

Definition 2.3. Let M be an m-dimensional topological manifold.

1. A chart on M is a pair (U,ϕ), where U ⊂M is open and ϕ : U −→ Vis a homeomorphism between U and an open subset V ⊂ Rm. Thecomponent functions ϕ1, ..., ϕm then are also called (local) coordinatesfor M on U ⊂M .

2. Two charts (Ui, ϕi), i = 1, 2, on a topological manifold M are called(C∞-)compatible if either U12 := U1 ∩ U2 is empty or the transitionmap (”coordinate change”)

ϕ2 ϕ−11 : ϕ1(U12) −→ ϕ2(U12)

is a diffeomorphism between the open sets ϕ1(U12) ⊂ V1 ⊂ Rm andϕ2(U12) ⊂ V2 ⊂ Rm.

3. A differentiable atlas A on a topological manifold M is a system

A = (Ui, ϕi); i ∈ I

of mutually (C∞-)compatible charts, such that M =⋃i∈I Ui.

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Example 2.4. 1. On the unit circle S1 the charts (Ui, ϕi := ψ−1i ), i = 1, 2

constitute a differentiable atlas: Indeed S1 = U1∪U2, and the transitionmap ϕ2 ϕ−1

1 :]− π, 0[∪]0, π[−→]0, π[∪]π, 2π[ looks as follows

ϕ2 ϕ−11 (x) =

x , if x ∈]0, π[x+ 2π , if x ∈]− π, 0[

.

2. The charts (Ui, σi), i = 1, 2 on Sn, cf. 2.2.3, constitute a differentiableatlas: Again we have Sn = U1 ∪ U2 and the transition map

σ2 σ−11 : Rn \ 0 −→ Rn \ 0, x 7→ x

||x||2.

3. Let W ⊂ Rn be an open subset and

F : W −→ Rn−m

a differentiable map, such that for all a ∈ M := F−1(0) ⊂ W theJacobian map

DF (a) : Rn −→ Rn−m

is surjective. For every point a ∈ M we shall construct a local chart(Ua, ϕa). We may assume that ∂F

∂(xm+1,...,xn)(a) 6= 0. Then the map

Φ : (x1, ..., xm, F1, ..., Fn−m) : Rn −→ Rn induces, according to theinverse function theorem, a diffeomorphism U −→ V between an openneighborhood U ⊂ W of a ∈ Rn and an open neighborhood V of(a1, ..., am, 0) ∈ Rn. As a consequence the map

ϕa : Ua := U ∩M −→ Va := y ∈ Rm; (y, 0) ∈ V ⊂ Rm × Rn−m,

x 7→ (x1, ..., xm)

is a homeomorphism. Then the collection A := (Ua, ϕa); a ∈ Mconstitutes a differentiable atlas on M . Note that all local coordinatesare obtained by choosing m suitable restrictions xi1|M , ..., xim |M of thecoordinate functions x1, ..., xn, i.e., with the choice of the set i1, ..., imdepending on the point a ∈M .

The last example shows that a differentiable atlas may depend on a lotof choices and can be unnecessarily big as well. So we need to say when twoatlases are ”equivalent”:

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Definition 2.5. 1. Two atlases A and A on an m-dimensional topologicalmanifold M are called equivalent if any chart in A is compatible withany chart in A.

2. A differentiable structure on a topological manifold is an equivalenceclass of differentiable atlases.

3. A differentiable manifold M is a topological manifold together with adifferentiable structure. We say that a differentiable atlas A is an at-las for the differentiable manifold M , if A defines (or belongs to) thedifferentiable structure of M .

If M is a differentiable manifold, then a ”chart (U,ϕ) on M” means alwaysa chart compatible with all the charts of a (resp. all) atlases defining thedifferentiable structure of M .

We leave the details of the following remark to the reader:

Remark 2.6. 1. Any open subset U ⊂ M of a differentiable manifoldinherits a natural differentiable structure.

2. The cartesian product M ×N of differentiable manifolds M,N carriesa natural differentiable structure.

Definition 2.7. Let M,N be differentiable manifolds of dimension m,n re-spectively.

1. A function f : M −→ R is differentiable if the functions f ϕ−1 :V −→ R are differentiable for all charts (U,ϕ : U −→ V ) ∈ A ina differentiable atlas for M . The same definition applies for mapsF : M −→ Rn. We denote

C∞(M) := f : M −→ R differentiable

the set of all differentiable functions, indeed a real vector space whichis even closed with respect to the multiplication of functions.

2. A continuous map F : M −→ N is called differentiable if all the mapsψ (F |F−1(W )) : F−1(W ) −→ Rn are differentiable, where (W,ψ) ∈ Bis any chart in an atlas B defining the differentiable structure of N .

3. A diffeomorphism F : M −→ N between two differentiable manifoldsM and N is a bijective differentiable map, such that its inverse F−1 :N −→M is differentiable as well.

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4. We say that M is diffeomorphic to N and write M ∼= N if there is adiffeomorphism F : M −→ N .

Note that differentiable functions are continuous, and that the definitionof differentiability is independent from the choice of the differentiable atlasesfor M and N .

Remark 2.8. Given a topological manifold M there are a lot of distinctdifferentiable structures, but the corresponding differentiable manifolds maybe diffeomorphic nevertheless: By a smooth type on M we mean the diffeo-morphism class of the differentiable manifold defined by some differentiablestructure on M .

1. For dimM ≤ 3 there is exactly one smooth type.

2. For dimM ≥ 4 it may happen that there is no differentiable structureon M at all.

3. A compact topological manifold of dimension at least 5 admits onlyfinitely many smooth types. E.g., the sphere Sn has the standardsmooth type as described above, but there may be other ones: For n =5, ..., 20 we obtain 1, 1, 28, 2, 8, 6, 992, 1, 3, 2, 16256, 2, 16, 16, 523264, 24smooth types respectively. For n = 4 it is not known whether thereare exotic smooth types, i.e. smooth types different from the standardsmooth type.

4. For n 6= 4 there is only the standard smooth type on Rn, while on R4

there are uncountably many different smooth types; some of them areobtained as follows: One considers an open subset U ⊂ R4 with thestandard smooth type; if there is a homeomorphism U ∼= R4 one getsan induced differentiable structure on R4.

Definition 2.9. Let M be an m-dimensional differentiable manifold. Asubset L ⊂ M is called a submanifold of codimension k iff for every pointa ∈ L there is a chart (U,ϕ), such that ϕ(U ∩L) = x = (x1, ...., xm) ∈ V :=ϕ(U);xm−k+1 = .... = xm = 0.

Note that a submanifold L ⊂M inherits from M a unique differentiablestructure, such that the inclusion L →M is differentiable.

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Remark 2.10. Complex Manifolds: We can literally apply the samestrategy as above with C replacing R. That leads to the notion of a complexstructure and complex manifolds. But there are also dramatic changes, sincethe condition for a map f : W −→ C on an open subset W ⊂ Cn to haveeverywhere a (complex) linear approximation, is an extremely strong one.Such functions are called holomorphic, we denote

O(M) := f : M −→ C holomorphic

the complex vector space of holomorphic functions. A holomorphic functionf ∈ O(M) has derivatives of any order, in fact it is even analytic, i.e. islocally described by its Taylor series.

Let us, without proof, mention the following difference: Given a compactset K ⊂ U contained in an open set U ⊂ M of a differentiable manifold M ,there is a function f ∈ C∞(M) with f |K ≡ 1 and f |M\U ≡ 0. On the otherhand, for a connected complex manifold M the following identity theoremholds: The restriction map

O(M) −→ O(W ), f 7→ f |Wis injective for any nonempty open subset W ⊂ M . Indeed if in additionM itself is compact, we have O(M) = C, i.e. there are only constant holo-morphic functions - this is an immediate consequence of the maximum prin-ciple for holomorphic functions. On the other hand Cm ∼= R2m and any

holomorphic map Cn ⊃ Wf−→ Cm is in particular differentiable as a map

R2n ⊃ Wf−→ R2m. So a complex manifold of dimension m can also be

regarded as a differentiable manifold of dimension 2m, the ”underlying dif-ferentiable manifold”.

3 Lie Groups

Definition 3.1. A (real) Lie group is a topological group G, which also car-ries the structure of a differentiable manifold, such that the group multiplica-tion G×G −→ G, (a, b) 7→ ab, as well as the inversion G −→ G, a 7→ a−1 aredifferentiable maps. If G even carries the structure of a complex manifoldand the group operations are holomorphic, we call G a complex Lie group.

Since real and complex Lie groups often can be treated simultaneouslywe shall from now on use the letter K in order to denote either R or C anduse the term K-Lie group in order to refer to real resp. complex Lie groups.

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Example 3.2. The general linear group

GLn(K) := A ∈ Kn,n; det(A) 6= 0

is a K-Lie group. Note that GL1(K) is nothing but the multiplicative groupK∗ := K \ 0 of K.

In the following we shall present a series of closed subgroupsG ⊂ GLn(K).In order to see that they are even Lie groups we use

Remark 3.3. Let G ⊂ GLn(K) be a closed subgroup, such that G = F−1(0)with a map F : GLn(K) −→ Km satisfying F (AX) = F (X) for all A ∈ G.If then DF (E) : Kn,n −→ Km is onto, the subgroup G ⊂ GLn(K) carriesa natural differentiable resp. complex structure, and with respect to thatstructure G is a K-Lie group. According to Example 2.4.3, it suffices tocheck that

DF (A) : Kn,n −→ Km

is onto for all A ∈ G. Denote λA : GLn(K) −→ GLn(K), X 7→ AX the leftmultiplication with A, a diffeomorphism. Since F = F λA, we obtain

DF (E) = DF (A) D(λA)(E) = DF (A) λA,

where we have used the fact that λA as a linear map coincides with its ownJacobian - here we denote also λA the map Kn,n −→ Kn,n, X 7→ AX. But Abeing invertible, λA : Kn,n −→ Kn,n is an isomorphism of vector spaces, sowith DF (E) the map DF (A) is surjective as well. The affine subspace

E + kerDF (E) ⊂ Kn,n

is the best approximation of F−1(0) = G at E by an affine subspace, it cannaturally be identified with the ”tangent space” TEG of G = F−1(0) at E,to be defined in the next chapter.

Now let us continue with our examples:

Example 3.4. 1. The special linear group

SLn(K) := A ∈ Kn,n; det(A) = 1

is a closed normal subgroup (as the kernel of the continuous homomor-phism det : GLn(K) −→ K∗). Take F (X) = det(X)− 1. Since

DF (E) = D(det)(E) = Tr

with the (surjective) trace map Tr : Kn,n −→ K,A = (αij) 7→∑n

i=1 αii,we can apply Remark 3.3. So SLn(K) is a K-Lie group.

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2. We consider a non degenerate bilinear form σ : Kn ×Kn −→ K, writeσ(x, y) = xTSy with a matrix S ∈ Kn,n. Then a matrix A ∈ GLn(K)preserves σ, i.e. σ(Ax,Ay) = σ(x, y) for all x, y ∈ Kn iff ATSA = S.Obviously the set of all such ”σ-isometries” forms a closed subgroup ofGLn(K). We look at the map

F : GLn(K) −→ Kn,n, X 7→ XTSX − S.

Then the Jacobian of F at E is

DF (E) : Kn,n −→ Kn,n, X 7→ XTS + SX.

Assume now S is either symmetric: ST = S or antisymmetric: ST =−S. Then F (GLn(K)) ⊂ Sn(K) resp. F (GLn(K)) ⊂ An(K), whereSn(K) denotes the vector space of all symmetric matrices and An(K)the vector space of all anti-symmetric matrices. Thus we may replacethe target of both F and DF (E) with Sn(K) resp. An(K). Sincea given matrix A ∈ Sn(K) resp. A ∈ An(K) is of the form A =DF (E)(X) with X := 1

2(S−1A), the Jacobian map of F at E is onto

and hence our isometry group even a Lie group. If we take S = E weobtain the K-orthogonal group

On(K) := A ∈ GLn(K);ATA = E,

while for even n = 2m and S =

(0 E−E 0

)the analogous group

Spn(K) := A ∈ GLn(K);ATSA = S

is called the K-symplectic group. We have

dimOn(K) = n2 − dimSn(K) =1

2n(n− 1)

and

dimSpn(K) = n2 − dimAn(K) =1

2n(n+ 1).

We remark that the real orthogonal group On(R) is compact, and thatdet(A) = ±1 for A ∈ On(K) as well as for A ∈ Spn(K).

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3. Now let us consider the hermitian form σ : Cn × Cn −→ C, σ(z, w) =zTw. The corresponding isometry group is the unitary group

U(n) := A ∈ GLn(C);ATA = E,

whileF : GLn(C) −→ Hn, X 7→ X

TX − E.

describes U(n) = F−1(0) as before. Here Hn ⊂ Cn,n denotes the realsubspace of all Hermitian matrices. The Jacobian map

DF (E) : Cn,n −→ Hn, X 7→ XT

+X

is again onto. Note that U(n) is not a complex Lie group, since F isnot holomorphic!

4. For G ⊂ GLn(K) let SG := G ∩ SLn(K). Since matrices A ∈ On(K)have determinant ±1, SOn(K) is an open subgroup of On(K) of index2, while Sp(n), as we hopefully shall see later on, is connected, hencenot only det(A) = ±1, but even det(A) = 1 for all A ∈ Spn(K).The group SU(n) ⊂ GLn(C) can be realized as follows: First notethat det(A) ∈ S1 for A ∈ U(n). Let W := GLn(C) \ det−1(R<0) andconsider the map

F : W −→ Hn × R, X 7→ (XTX − E, arg(det(X)))

where, say, −π < arg(.) < π, with SU(n) = F−1(0) and Jacobian

DF (E) : Cn,n −→ Hn × R, X 7→ (XT

+X, Im(Tr(X))),

which is onto, since (A, λ) ∈ Hn × R has X = 12(A + in−1λE) as an

inverse image.

5. Let V0 := 0 ⊂ V1 ⊂ ... ⊂ Vn = Kn be an increasing sequence ofsubspaces (a ”flag”). If Vi = Ke1 + ...+Kei the subgroup

UTn(K) := A ∈ GLn(K);A(Vi) ⊂ Vi

is a K-Lie group, consisting of the invertible upper triangular matrices,indeed the underlying differentiable or complex manifold is nothing but(K∗)n ×Kn(n−1)/2 and we may argue as in the case of GLn(K).

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There is a canonical homomorphism

UTn(K) −→ GL(Vn/Vn−1)× ...×GL(V2/V1)×GL(V1) ∼= (K∗)n,

its kernel UUn(K) ⊂ UTn is K-Lie group, it consists of all upper trian-gular matrices with diagonal entries equal to 1 (”unipotent” matrices).

All our above Lie groups are closed subgroups of GLn(K). The belowtheorem tells us that any such group is indeed a Lie group:

Theorem 3.5. Let H ⊂ G be a closed subgroup of the real Lie group G.Then H ⊂ G is a closed submanifold of G and in particular again a Liegroup.

But since we are far from being able to prove it, we have preferred to givein any particular case a complete argument. – For any topological groupG the connected component G0 ⊂ G of G containing the neutral elemente ∈ G is a normal subgroup, the group operations being continuous. Since aLie group G as a topological manifold is locally connected, G0 is even open,indeed the minimal open subgroup of G, a connected Lie group G beinggenerated by any open neighborhood of e ∈ G.

Remark 3.6. A connected Lie group G is countable at infinity: Take acompact neighborhood K ⊂ G of the neutral element, w.l.o.g. K = K−1 :=a−1; a ∈ K. ThenG is the ascending unionG =

⋃∞n=1 K

n with the compactsubsets Kn := a1 · ... · an; a1, ..., an ∈ K: The right hand side is obviouslyan open subgroup of G, hence equals G, the group G being connected.

4 Vector Fields

On a differentiable manifold M there is no natural notion of a derivative of afunction f ∈ C∞(M), since in order to differentiate we need local coordinates.We are now going to define objects with respect to which functions can bedifferentiated in a point a ∈M or globally.

Definition 4.1. A tangent vector Xa at a point a ∈ M of a differentiablemanifold M is a linear map Xa : C∞(M) −→ R satisfying the followingLeibniz rule:

Xa(fg) = f(a)Xa(g) +Xa(f)g(a).

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The set of all tangent vectors of M at a ∈ M forms a vector space TaM ,called the tangent space of M at a ∈M .

Remark 4.2. (1) We have Xa(R) = 0 for every tangent vector Xa ∈ TaM ,since Xa(1) = Xa(1

2) = Xa(1) +Xa(1).

(2) Take a chart ϕ : U → V ⊂ Rn with a ∈ U and ϕ(a) = 0. Then themaps

∂ai := ∂ϕ,ai : f 7→ ∂f ϕ−1

∂xi(0), i = 1, ..., n,

are tangent vectors at a.

(3) Another, may be more geometric, construction that avoids the choice ofcharts is the following: To any curve, i.e. differentiable map, γ : I →Mdefined on an open interval I ⊂ R with γ(t0) = a for some t0 ∈ I wecan associate the tangent vector γ(t0) ∈ TaM defined by

γ(t0) : f 7→ (f γ)′(t0) .

The vector γ(t0) is called the tangent vector of the curve γ : I → Mat t0 ∈ I. In particular the tangent space TaRn is naturally isomorphicto Rn itself: associate to x ∈ Rn the tangent vector γx(0) with thecurve γx(t) := a+ tx. The adjective ”natural” means here that it onlydepends on Rn as vector space, not on the choice of a particular base(e.g. the standard base) of Rn.

Theorem 4.3. Using the notation of Remark 4.2.2 we have

TaM =n⊕i=1

R · ∂ai ,

i.e. the tangent vectors ∂ai := ∂ϕ,ai form a basis of the tangent space TaM .

For the proof we need

Lemma 4.4. 1. If f ∈ C∞(M) vanishes in a neighborhood of a ∈ M ,then Xa(f) = 0 for all tangent vectors Xa ∈ TaM .

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2. Let U ⊂ M be open. Denote % : C∞(M) −→ C∞(U), f 7→ f |U therestriction from M to U . Then the map

TaU −→ TaM,Xa 7→ Xa %,

is an isomorphism.

Proof. 1.) If f vanishes near a, take a function g ∈ C∞(M) with g = 1 neara and fg = 0. Then 0 = Xa(fg) = g(a)Xa(f) + f(a)Xa(g) = Xa(f).2.) Injectivity: Assume Xa % = 0. Take any function f ∈ C∞(U). Choosef ∈ C∞(M) with f = f near a. Then, according to the first part, we haveXa(f) = Xa(f |U) = 0. Now the function f ∈ C∞(U) being arbitrary, weobtain Xa = 0.Surjectivity: For Ya ∈ TaM define Xa ∈ TaU by its value on f ∈ C∞(U) asfollows

Xa(f) := Ya(f),

where again f ∈ C∞(M) with f = f near a. Then Xa(f) is well defined asa consequence of the first part and obviously Xa % = Ya.

Proof of 4.3. As a consequence of 4.4 we may assume, with the notation ofRem.4.2.2, M = U = V ⊂ Rn and show that the tangent vectors ∂ 0

i ∈ T0Vwith ∂ 0

i (f) := ∂f∂xi

(0) form a basis of T0V . Since ∂ 0i (xj) = δij they are linearly

independent. On the other hand, for any X0 ∈ T0V we have

X0 =n∑i=1

X0(xi)∂0i .

Take f ∈ C∞(V ). After, may be, a shrinking of V we may assume f = f(a)+∑ni=1 xifi with fi ∈ C∞(V ) and then obtain Xa(f) =

∑ni=1 Xa(xi)fi(a) =∑n

i=1Xa(xi)∂ai (f).

Tangent vectors on a complex manifold (Only for readers feeling athome in complex analysis!): For a complex manifold M we have to modifythe definition of a tangent vector, since O(M) may be too small. Instead ofworking with O(M) we have to introduce the concept of a germ of a functionnear a ∈ M . Let us do that simultaneously for differentiable and complexmanifolds: We consider pairs (U, f), where f : U −→ K is any K-valuedfunction defined on an open neighborhood U of a ∈M . A germ of a function

17

is an equivalence class of such pairs, where two pairs (U, f) and (V, g) areequivalent if there is an open neighborhood W ⊂ U ∩V of a with f |W = g|W .Given f : U −→ K the corresponding germ is denoted fa. We leave it to thereader to define the sum and product of germs. Then we obtain the vectorspaces C∞a and Oa of germs of differentiable resp. holomorphic functionsnear a ∈M . It follows from the above remark 4.2.2, that any tangent vectorXa : C∞(M) −→ R factorizes uniquely through a linear map C∞a −→ Rsatisfying the Leibniz rule, and on the other hand, every such map yields atangent vector by composition with the natural (surjective!) map

C∞(M) −→ C∞a , f 7→ fa,

associating to a function f ∈ C∞(M) its germ at a ∈M . Since for a complexmanifold the corresponding map

O(M) −→ Oa, f 7→ fa,

is never surjective for dimM > 0, we have to define a complex tangentvector at a ∈M as a C-linear map Za : Oa −→ C satisfying the Leibniz rule.Again we obtain a basis ∂a1 , ..., ∂

an of TaM given by complex differentiation

with respect to local complex coordinates. Furthermore, given a holomorphic”curve”, i.e. a holomorphic map γ : G −→ M defined on an open subsetG ⊂ C we may define its tangent vector γ(z0) for any z0 ∈ G literally as inthe real case.

Given a complex manifold M , denote MR the underlying differentiablemanifold. Then there is a natural identification of real tangent vectors at a,i.e. tangent vectors of the differentiable manifold MR, and complex tangentvectors: More precisely, for a ∈ M , there is a natural isomorphism of realvector spaces (Only the target is even a complex vector space!)

Ta(MR) −→ TaM,Xa 7→ Za := XCa |Oa ,

whereXCa := Xa + iXa : C∞,Ca = C∞a ⊕ iC∞a −→ C = R⊕ iR

with the vector space C∞,Ca = C∞a ⊕ iC∞a of germs of complex valued dif-ferentiable functions near a ∈ M . To be less mysterious and more explicit:Given complex local coordinates z1 = x1 + iy1, ..., zn = xn + iyn near a, theabove map acts as follows

∂ a

∂xk7→ ∂ a

∂zk,∂ a

∂yk7→ i

∂ a

∂zk,

18

since the above derivations coincide on Oa ⊂ C∞,Ca .

Differentiable maps induce linear maps between tangent spaces:

Definition 4.5. Given a differentiable map F : M → N between the differ-entiable manifolds M and N , there is an induced homomorphism of tangentspaces:

F∗ := TaF : TaM → TF (a)N

defined byF∗(Xa) : C∞(N)→ R, f 7→ Xa(f F ) .

It is called the tangent map of F at a ∈M .

Obviously we have for a curve γ : (−ε, ε)→M with γ(0) = a that

F∗(γ(0)) = δ(0), where δ := F γ .

For explicit computations we note that, if F = (F1, ..., Fm) : U → W is adifferentiable map between the open sets U ⊂ Rn and W ⊂ Rm, and b = F (a)for a ∈ U , then with respect to the bases ∂a1 , ..., ∂

an of TaU and ∂b1, ..., ∂

bm of

TbW the linear map TaF has the matrix:

DF (a) =

(∂Fi∂xj

(a)

)1≤i≤m,1≤j≤n

∈ Rm,n ,

the Jacobi matrix of F at a ∈ U .Furthermore it is immediate from the definition, that the tangent map

behaves functorially, i.e. if F : M1 →M2 and G : M2 →M3 are differentiablemaps, then G F : M1 →M3 is again differentiable and the chain rule

Ta(G F ) = TF (a)G TaF

holds.All the tangent vectors at points in a differentiable n-manifold M form a

differentiable n2-manifold:

Definition 4.6. Let M be a differentiable n-manifold. The tangent bundleTM is, as a set, the disjoint union

TM :=⋃a∈M

TaM

19

of all tangent spaces at points a ∈M . Denote π : TM →M the map, whichassociates to a tangent vector Xa ∈ TaM its “base point” a ∈M . Now given

a chart ϕ : U∼=→ V ⊂ Rn on M , we consider the bijective map (trivialization)

Tϕ := (ϕ π, ϕ∗) : π−1(U)→ V × Rn,

where we use the natural isomorphism TaRn ∼= Rn as explained above andϕ∗|TaM = Taϕ. We endow TM with a topology: a set W ⊂ TM is open ifTϕ(W ∩ U) ⊂ Rn × Rn is open for all charts (U,ϕ) in an atlas A for M .Finally, the charts (π−1(U), Tϕ) with (U,ϕ) ∈ A define an atlas on TM .

In order to see that this idea works we need to understand the coordinatechanges for two charts (π−1(U), Tϕ) and (π−1(U), T ϕ) of TM : We mayassume U = U and then obtain with F := ϕ ϕ−1 : V −→ V the followingcoordinate change on the tangent level:

T ϕ (Tϕ)−1 : V × Rn −→ V × Rn, (x, y) 7→ (F (x), DF (x)y).

Now we can generalize Definition 4.5: given a differentiable map F :M → N the pointwise tangent maps TaF : TaM → TF (a)N combine to adifferentiable map TF : TM → TN , i.e.

TF |TaM := TaF : TaM → TF (a)N.

Indeed, the map TF fits into a commutative diagram

TMTF−→ TN

↓ ↓M

F−→ N

,

i.e. πN TF = F πM holds with the projections πM : TM −→ M andπN : TN → N of the respective tangent bundles.

Remark 4.7. For a complex manifold M the above construction works aswell, leading to a complex manifold TM , its holomorphic tangent bundle. Onthe other hand, we already have seen that for a complex manifold M , thereis a natural isomorphism

Ta(MR) ∼= TaM

20

for every point a ∈ M . Since that isomorphism depends ”differentiably” onthe base point a ∈M , there is a natural identification

T (MR) = TM

of differentiable manifolds.

Definition 4.8. A (holomorphic) vector field X on an open subset U ⊂ Mof a differentiable (complex) manifold M is a differentiable (holomorphic)section of the projection π : TM → M , i.e., a differentiable (holomorphic)map

X : U → TM

satisfying π X = idU , with other words, X(a) ∈ TaM for all a ∈ U . In thatcase we also write Xa := X(a). We denote Θ(U) the set of all (holomorphic)vector fields on U ⊂M .

Remark 4.9. (1) The set Θ(U) carries, with the argument-wise algebraicoperations, in a natural way the structure of a real vector space. Infact the scalar multiplication

R×Θ(U)→ Θ(U)

can be extended to a multiplication by functions:

C∞(U)×Θ(U)→ Θ(U), (f,X) 7→ fX,

where

(fX)a := f(a)Xa .

That follows immediately from the fact that the maps Taϕ : TaM −→Rn are linear isomorphisms.

(2) Let (U,ϕ) be a chart. Then

∂i := ∂ϕi : U −→ TM, a 7→ ∂ai for i = 1, ..., n,

with

∂ai (f) :=∂f ϕ−1

∂xi(ϕ(a))

21

are vector fields on U , the ”coordinate vector fields” associated to thelocal chart (or local coordinates) xi = ϕi(a), i = 1, ..., n. Since TaM =⊕n

i=1 R∂ai , any section X : U −→ TM of π : TM −→M can be written

X =n∑i=1

gi∂i,

with functions gi : U −→ R, and we have

X ∈ Θ(U)⇐⇒ g1, ..., gn ∈ C∞(U).

(3) Note that on an arbitrary differentiable manifold M it is in general notpossible to find vector fieldsX1, ..., Xn ∈ Θ(M), such that (X1)a, ..., (Xn)ais a frame at a, i.e., a basis of TaM , for all a ∈M . If such vector fieldsexist, the manifold M is called parallelizable. So the open subsets U ,where a local chart ϕ : U −→ V is defined, are always parallelizable.As we shall see in the next section the underlying manifold of a Liegroup is always parallelizable.

(4) The vector fields onM can be identified with derivationsD : C∞(M)→C∞(M), i.e. linear maps satisfying the Leibniz rule D(fg) = D(f)g +fD(g) for all f, g ∈ C∞(M). Given a vector field X ∈ Θ(M) thecorresponding derivation X : C∞(M)→ C∞(M), f 7→ X(f) is definedby (X(f))(a) := Xa(f). In fact, every derivation D : C∞(M) →C∞(M) is obtained from a vector field: Take X ∈ Θ(M) with

Xa : C∞(M)→ R, f 7→ D(f)(a) .

(5) For an open subset U ⊂ M the tangent bundle TU is identified, in anatural way, with the open subset π−1(U) ⊂ TM .

(6) Let F : M → N be a differentiable map. Given a vector field X ∈Θ(M), we can consider TF X : M → TN , but that map does notin general factor through N , e.g. if F is not injective. But it does ifF : M → N is a diffeomorphism: then we may define a map

F∗ : Θ(M)→ Θ(N), X 7→ F∗(X) := TF X F−1 ∈ Θ(N),

the push forward of vector fields with respect to a diffeomorphism.

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The vector space Θ(M) carries a further algebraic structure: though thecompositions XY and Y X of two derivations X, Y : C∞(M)→ C∞(M) areno longer derivations, their commutator is:

(XY − Y X)fg = XY (fg)− Y X(fg)

= X(fY g + gY f)− Y (fXg + gXf)

= fXY g + (Xf)Y g + gXY f + (Xg)Y f

− fY Xg − (Y f)(Xg)− gY Xf − (Y g)(Xf)

= fXY g − fY Xg + gXY f − gY Xf= f(XY − Y X)g + g(XY − Y X)f.

Definition 4.10. The Lie bracket [X, Y ] ∈ Θ(M) of two vector fields X, Y ∈Θ(M) is the commutator of the derivations X, Y : C∞(M)→ C∞(M), i.e.

[X, Y ] := XY − Y X,

or, in other words, the vector field [X, Y ] satisfying

[X, Y ]a(f) := Xa(Y (f))− Ya(X(f))

for all differentiable functions f ∈ C∞(M) at every point a ∈M .

Note that the tangent vector [X, Y ]a is not a function of the valuesXa, Ya ∈ TaM only, since the local behavior of the vector fields X, Y neara ∈M also enters in the computation rule. If x1, ..., xn are local coordinateson U ⊂M , and X, Y ∈ Θ(U) have representations

X =n∑i=1

fi ∂i, Y =n∑i=1

gi ∂i,

then

[X, Y ] =n∑i=1

(X(gi)− Y (fi)) ∂i .

So, in particular, [∂i, ∂j] = 0 for coordinate vector fields. On the other handwe mention:

Theorem 4.11. (Frobenius Theorem) Let X1, ..., Xn ∈ Θ(M) be pair-wise commuting vector fields, i.e. [Xi, Xj] = 0 for 1 ≤ i, j ≤ n. Then every

23

point a ∈ M , such that (X1)a, ..., (Xn)a is a frame at a (i.e. a basis of thetangent space TaM) admits a neighborhood U ⊂ M with local coordinatesx1, ..., xn ∈ C∞(U) such that

Xi|U = ∂i.

The proof relies on the fact that given the flows (µt) and (µt) of commut-ing vector fields X, X ∈ Θ(M), one has µs µt = µt µs for all sufficientlysmall s, t ∈ R.

The Lie bracket defines on the vector space Θ(M) the structure of a Liealgebra, a notion which plays a key role in the investigation of Lie groups:

Definition 4.12. A Lie algebra g over K is a K-vector space endowed withan antisymmetric (or alternating) bilinear map [.., ..] : g× g→ g, i.e.

[X, Y ] = −[Y,X], ∀ X, Y ∈ g, in particular [X,X] = 0,

such that the Jacobi identity holds:

[X, [Y, Z]] + [Y, [Z,X]] + [Z, [X, Y ]] = 0.

We shall need later on

Proposition 4.13. Let F : M −→ N be a diffeomorphism resp. a biholo-morphic map. Then the push forward of vector fields F∗ : Θ(M) −→ Θ(N)is an isomorphism of Lie algebras.

Proof. Regarding X ∈ Θ(M) as a derivation X : C∞(M) −→ C∞(M) thederivation F∗(X) : C∞(N) −→ C∞(N) satisfies

F∗(X)g = X(g F ) F−1.

HenceF∗([X, Y ])g = XY (g F ) F−1 − Y X(g F ) F−1,

while

F∗(X)F∗(Y )g = F∗(X)(Y (g F ) F−1) = XY (g F ) F−1

and in the same way

F∗(Y )F∗(X)g = Y X(g F ) F−1.

This implies our claim.

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Definition 4.14. Let X ∈ Θ(M) be a vector field. A smooth curve γ : I →M resp. a holomorphic curve γ : G→M defined on an open interval I ⊂ Rresp. on an open domain G ⊂ C is called an integral curve of the vector fieldX, if γ(t) = Xγ(t) for all t ∈ I resp. t ∈ G.

Remark 4.15. The basic theorem in the theory of ordinary differential equa-tions says that, given a vector field X ∈ Θ(M) on a differentiable manifold Mand a point a ∈M , there is an integral curve γ : I →M defined on an openinterval I 3 0 such that γ(0) = a, and if γ : I → M is a second such curve,then γ and γ coincide on the intersection I ∩ I. If M is compact, we canalways assume I = R. Moreover, there is a differentiable map µ : U → Mdefined on an open neighborhood U ⊂ M × R of M × 0 → M × R, suchthat U ∩ (x × R) is an interval for all x ∈ M and t → µ(x, t) an integralcurve of X satisfying µ(x, 0) = x. If, moreover, M is compact, this map isdefined on all of M × R. The flow of the vector field X then is the family(µt)t∈R of the differentiable maps

µt : M →M,x 7→ µ(x, t).

In fact, we haveµ0 = idM , µs+t = µs µt,

since integral curves are uniquely determined by their initial values and t 7→γ(s+ t) is an integral curve with the value γ(s) at t = 0. Since µ0 = idM , itfollows that each map µt : M →M is a diffeomorphism with inverse µ−t.

5 The Lie Algebra of a Lie Group

For a Lie group G there are distinguished vector fields: Given an elementa ∈ G we denote λa : G −→ G, x 7→ ax, resp. %a : G −→ G, x 7→ xa, the leftresp. right multiplication (”left resp. right translation”) with a ∈ G. Theydefine diffeomorphisms (resp. biholomorphic maps) G −→ G and induce Liealgebra isomorphisms

(λa)∗, (%a)∗ : Θ(G) −→ Θ(G).

Definition 5.1. A vector field X ∈ Θ(G) on a Lie group G is called leftresp. right invariant if (λa)∗(X) = X resp. (%a)∗(X) = X for all g ∈ G.We denote Lie(G) ⊂ Θ(G) or simply g := Lie(G) the subspace of all leftinvariant vector fields.

25

In the following we usually consider only left invariant vector fields, buteverything holds - mutatis mutandis - for right invariant vector fields as well.

Remark 5.2. 1. The subspace g ⊂ Θ(G) is a Lie subalgebra, i.e. closedwith respect to the Lie bracket:

[g, g] ⊂ g.

This follows immediately from the fact that the (λa)∗ : Θ(G) −→ Θ(G)are Lie group homomorphisms.

2. A vector field X ∈ Θ(G) is left invariant iff Xa = Te(λa)(Xe) for alla ∈ G. As a consequence the evaluation map

g −→ TeG,X 7→ Xe

is an isomorphism of vector spaces. Hence g is a finite dimensional vec-tor space with dim g = dimG (where dimG denotes the dimension ofG as differentiable or complex manifold). Indeed, usually one identifiesg with TeG via the above isomorphism.

Example 5.3. Let us consider G = GLn(K) and denote Ξ := (ξij) a variablematrix in GLn(K). Furthermore we need the following matrix

∆ := (∂ij),

the entries of which are the coordinate vector fields with respect to the co-ordinates ξij. To simplify notation we introduce a scalar product

A ·B :=∑i,j

AijBij

for matrices A = (Aij), B = (Bij). Since TEλC = λC , the left invariant vectorfield X with XE = A ·∆E =

∑i,j Aij∂

Eij looks as follows

X = (ΞA) ·∆ =∑i,j

(ΞA)ij∂ij.

Let us now compute the Lie bracket of vector fields X = (ΞA) · ∆, Y =(ΞB) ·∆. We already know that

[X, Y ] = (X(ΞB)− Y (ΞA)) ·∆,

26

where X and Y apply to each entry of the matrices ΞB and ΞA separately.Now

X(ΞB) = X(Ξ)B

and

∂ijΞ = E(i, j)

with the matrix E(i, j) such that E(i, j)k` = δkiδ`j. Thus, using

X(Ξ) = ((ΞA) ·∆) (Ξ) =∑i,j

(ΞA)ij∂ij(Ξ) =∑i,j

(ΞA)ijE(i, j) = ΞA,

we obtain

X(ΞB) = ΞAB.

Since by symmetry

Y (ΞA) = ΞBA,

we finally arrive at

[X, Y ] = (Ξ [A,B]) ·∆

with the commutator

[A,B] = AB −BA

of the matrices A,B ∈ Kn,n.

Definition 5.4. A homomorphism ϕ : G −→ H between Lie groups G,His a differentiable resp. holomorphic group homomorphism. We denoteHom(G,H) the set of all such homomorphisms.

Remark 5.5. The kernel and image of a homomorphism ϕ : G −→ H of Liegroups:

1. The kernel ker(ϕ) ⊂ G is clearly a closed subgroup. Indeed it is easilyseen to be also a submanifold: The tangent map Taϕ : TaG −→ Tϕ(a)Hhas for all a ∈ G the same rank, and as a consequence of that, thereis for every a ∈ G an open neighborhood U ⊂ G of a and W ⊂ Hof ϕ(a), such that with respect to suitable local coordinates x1, ..., xnon U and y1, ..., ym on W the map ϕ takes the form (x1, ..., xn) 7→(x1, ..., xk, 0, ...0) with some k ≤ n.

27

2. The image ϕ(G) ⊂ H need neither be a closed subgroup of H nor atopological manifold with respect to the relative topology as a subspaceof H. As an example consider the injective homomorphism ϕ : R −→S1 × S1, t 7→ (eit, ei

√2t) – the real line wound up around a life belt.

But if we refine the relative topology then it is: Define a base for thetopology of ϕ(G) to consist of the sets ϕ(U) with U ⊂ G as above. Thedifferentiable structure now is inherited from H, the sets ϕ(U) beingsubmanifolds of H. So there is a unique Lie group structure on ϕ(G)!On the other hand, if ϕ(G) ⊂ H is closed and G connected, then bothLie group structures coincide, that one as a closed subgroup of H andthat as an image of a homomorphism of Lie groups. In order to seethat one uses the fact that G is countable at infinity and that an opensubset of Rm never is the countable union of closed submanifolds ofdimension < m (use e.g. measure theory - a compact part of such asubmanifold is a Lebesgue zero set in Rm.).

Proposition 5.6. Any homomorphism of Lie groups ϕ : G −→ H induces ahomomorphism of the corresponding Lie algebras dϕ : g −→ h.

Proof. The homomorphism dϕ is defined as g ∼= TeGTeϕ−→ TeH ∼= h, and we

have to show that dϕ preserves the Lie bracket. We may assume that bothG and H are connected and then consider separately G −→ ϕ(G) (resp. thecase of a surjective homomorphism) and the inclusion ϕ(G) −→ H resp. aninjective homomorphism.

The first case is easy: Since ϕ : G −→ H is onto, the pull back ϕ∗ :C∞(H) −→ C∞(G) is injective. Now g can be regarded as the set of allderivations D : C∞(G) −→ C∞(G) commuting with left translation, i.e.D(f λa) = (Df) λa and dϕ is nothing but the restriction D 7→ D|C∞(H).The claim follows, since the Lie bracket is nothing but the commutator of twoderivations. But note that a general derivation does not satisfy D(C∞(H)) ⊂C∞(H).

The case of an injective homomorphism (resp. an inclusion) ϕ : G −→ H:Given X ∈ g the vector field X := dϕ(X) ∈ h is the unique left invariantextension of X, i.e. X|G = X. Now the claim follows from the equality

[X, Y ]|G = [X, Y ],

the left hand side being a left invariant vector field. It is easily checked usinglocal coordinates x1, ...., xn with G ⊂ H being defined by xk+1 = ... = xn = 0.

28

If we write

X =n∑i=1

fi∂i, Y =n∑i=1

gi∂i,

then gi|G = 0 = fi|G for i ≥ k + 1. Hence

[X, Y ] =n∑i=1

(n∑j=1

fj∂jgi − gj∂jfi

)∂i

and

[X, Y ]|G =k∑i=1

(k∑j=1

fj∂jgi − gj∂jfi

)∂i = [X, Y ],

since we have gj|G = 0 = fj|G for j ≥ k + 1 and ∂jgi = 0 = ∂jfi forj ≤ k, i ≥ k + 1.

Example 5.7. Let us discuss the Lie algebras g = Lie(G) for the Lie groupsof Example 3.3. There G ⊂ GLn(K) is a closed subgroup of GLn(K).Then g ⊂ gln(K) = Kn,n is a subalgebra, the kernel of the map DF (E) :gln(K) −→ Km. We obtain:

1. sln(K) = A ∈ Kn,n; Tr(A) = 0.

2. If G ⊂ GLn(K) is the isometry group of a the bilinear form σ(x, y) =xTSy, then g = A ∈ Kn,n;ATS + SA = 0, in particular

son(K) = A ∈ Kn,n;AT + A = 0

consists of the skew symmetric matrices.

3. un = A ∈ Cn,n;AT

= −A.

4. sun = A ∈ Cn,n;AT

= −A,Tr(A) = 0.

5. utn(K) := Lie(UTn(K)) consists of all upper triangular matrices and

6. uun(K) := Lie(UUn(K)) of all upper triangular matrices with onlyzeros on the diagonal.

29

6 Homogeneous Spaces

Let H ⊂ G be a closed subgroup of the Lie group G. We want to explain howto make the left coset space B := G/H a differentiable manifold. Denote π :G −→ B, g 7→ gH, the quotient projection. We endow B with the π-quotienttopology. Then π becomes an open map, since π−1(π(V )) =

⋃h∈H V h.

Now let us show that B is Hausdorff: Consider two different points x =aH, y = bH. Equivalently (a, b) ∈ G×G\L with the closed set L = ψ−1(H),where ψ : G × G −→ G, (ξ, η) 7→ ξ−1η, is continuous. Hence L ⊂ G × Gis closed, and there is a neighbourhood U × V ⊂ G × G \ L of (a, b) withopen U, V ⊂ G. Then π(U) and π(V ) are disjoint open neighbourhoods ofthe points x, y ∈ G.

Indeed L ⊂ G × G is even a submanifold as the inverse image of a sub-manifold with respect to a submersion: A differentiable map f : M −→ N iscalled a submersion, if all its tangent maps Taf : TaM −→ Tf(a)N are onto(:=surjective). Since

gψ(ξ, η)g = ψ(ξg−1, ηg),

and (ξ, η) 7→ (ξg−1, ηg) is a diffeomorphism G × G −→ G × G as well asζ 7→ gζg a diffeomorphism G −→ G, it suffices to check that

T(e,e)ψ : T(e,e)(G×G) ∼= TeG⊕ TeG −→ TeG

is onto; indeed T(e,e)ψ(Xe, Ye) = Ye − Xe. This follows from the fact thatthe group law µ : G×G −→ G, (ξ, η) 7→ ξη has differential T(e,e)µ(Xe, Ye) =Xe + Ye as a consequence of µ(., e) = idG = µ(e, .). It follows immediatelythat the inversion ι : G −→ G, ξ 7→ ξ−1 has differential Teι(Xe) = −Xe.

Our next aim is to construct on a suitable neighbourhood U of x0 := eHa continuous section σ : U −→ G of the quotient projection π, i.e., suchthat π σ := idU . As candidate for σ(U) we choose a slice S ⊂ G to thesubmanifoldH at e, i.e. S ⊂ G is a submanifold, e ∈ S and TeG = TeS⊕TeH.If G has dimension n and H codimension ` := dimG− dimH, then we maytake S ∼= B` with an open ball B` ⊂ K`. The main point now is to provethat after a shrinking of S the image U := π(S) is open in B and the map

τ : S ×H −→ π−1(U), (s, h) 7→ sh

diffeomorphic resp. biholomorphic. First of all, the map σ is diffeomorphic(biholomorphic) near (e, e). If σ : V1 × V2 −→ W 3 e has that property,replace S with V1; then we have π−1(U) =

⋃h∈HWh, so U := π(S) is open

30

and our map τ : S × H −→ π−1(U) a surjective locally diffeomorphic resp.biholomorphic map (since π(s, h) = π(s, e)h). It remains to show that it isinjective. That is true if π|S is injective resp.

(S × S) ∩ L = ∆ := (s, s); s ∈ S.

If we can prove that (S × S) ∩ L is an `-dimensional submanifold, we aredone, since ∆ ⊂ (S × S) ∩ L is as well, hence not only closed, but also openin (S × S) ∩ L, i.e. it is the connected component of (S × S) ∩ L containing(e, e). Thus a shrinking of S leads to the desired equality (S × S) ∩ L = ∆.

In order to see that (S×S)∩L is, after some shrinking, an `-dimensionalmanifold, we note that

T(e,e)(G×G) = T(e,e)(S × S) + T(e,e)L,

with the RHS being nothing but

TeS × TeS + (TeH × 0+ ∆TeG×TeG)

= TeG× TeS + ∆TeG×TeG = TeG× TeG,

and apply

Proposition 6.1. Let M be a differentiable manifold and a ∈ L∩N a pointin the intersection of two submanifolds L,N ⊂M . If TaM = TaL+TaN theintersection L ∩ N is near a a submanifold of dimension dimN + dimL −dimM .

Proof. Let n = dimN, ` = dimL,m = dimM . We may assume

N = F−1(0)

with a submersion F : U −→ Rm−n defined on a neighbourhood U of a ∈M .Since TaN = ker(TaF ) our assumption TaM = TaL+ TaN implies that evenTa(F |(L∩U)) is onto. Hence after a shrinking of U the restriction F |L∩U is asubmersion as well, in particular L∩N ∩U = (F |U∩L)−1(0) is a submanifoldof dimension `− (m− n).

In order to write down explicitly a differentiable atlas for B = G/H weuse that B carries a natural (left) G-action:

G×B −→ B, (g, x = aH) 7→ gx := (ga)H.

31

Now, for a ∈ G take Ua := aU ⊂ B, as local coordinates on Ua consider themap

ϕa : Ua −→ B`, x 7→ F (σ(a−1x)),

where F : S −→ B` is a fixed diffeomorphism (biholomorphic map) with aball B` ⊂ K`..

Corollary 6.2. Let H ⊂ G be a closed normal K-Lie subgroup of the K-Lie group G. Then G/H is again a K-Lie group with the quotient mapG −→ G/H being a homomorphism of K-Lie groups.

The map π : G −→ B has a remarkable geometric structure: It is anH-principal bundle:

Definition 6.3. Let H be a Lie group. An H-principal bundle consists of

1. a differentiable/holomorphic right action

E ×H −→ E

on a differentiable/complex manifold E, also called the total space ofthe bundle

2. and the bundle projection

π : E −→ B,

a surjective differentiable/holomorphic H-invariant map to a differen-tiable or complex manifold B – the base (space) of the bundle – , suchthat every point b ∈ B admits an open neighbourhood U ⊂ B, overwhich there is an H-equivariant trivialization of the bundle projection

τ : U ×H −→ π−1(U),

i.e. a diffeomorphism/biholomorphic map satisfying

π τ = prU : U ×H −→ U

andτ(x, hh) = τ(x, h)h

for all x ∈ U, h, h ∈ H, i.e. τ transforms the right action of H onU ×H by right translation on the second factor into the right action ofH on E.

32

Remark 6.4. Given an H-principal bundle π : E −→ B there is an opencover B =

⋃i∈I Ui together with equivariant trivializations

τi : Ui ×H −→ π−1(Ui).

The bundle π : E −→ B then can be reconstructed from that cover and thetransition functions

fij : Uij := Ui ∩ Uj −→ H,

defined by the condition that the composition

Uij ×Hτi−→ π−1(Uij)

τ−1j−→ Uij ×H

satisfies

(x, e) 7→ (x, fij(x)).

Since τ−1j τi is H-equivariant, it then takes the form

τ−1j τi : (x, h) 7→ (x, fij(x)h).

Note that the bundle E −→ B can (up to a bundle isomorphism) be recon-structed from the cover (Ui)i∈I and the functions fij.

Indeed, whenever there is an open cover B =⋃i∈I Ui together with func-

tions fij : Uij −→ H for all (i, j) ∈ I2 satisfying the cocycle condition:

fik = fjkfij

on Uijk = Ui ∩ Uj ∩ Uk for all (i, j, k) ∈ I3, we may construct an associatedH-principal bundle E −→ B as follows:

E =

(⋃i∈I

Ui ×H

)/∼ ,

where the union is regarded as a disjoint union and the equivalence relationidentifies as follows:

Ui ×H ⊃ Uij ×H 3 (x, h) ∼ (x, fij(x)h) ∈ Uij ×H ⊂ Uj ×H.

33

Example 6.5. Our left coset map π : E := G −→ B := G/H is an H-principal bundle: First of all

G/H =⋃a∈G

Ua

with Ua := aU . As next define the section σa : Ua −→ E = G of the bundleprojection π : E −→ B by σa(x) := aσ(a−1x), and finally

τa : Ua ×H −→ π−1(Ua), (x, h) 7→ σa(x)h.

The corresponding function fa,b : Ua,b = Ua ∩ Ub −→ H then turns out to be

fa,b(x) = σb(x)−1σa(x).

7 The Exponential Map

In order to study the geometry of a K-Lie group one investigates its ”oneparameter subgroups”, i.e. the homomorphisms γ : K −→ G from theadditive group K to G.

Theorem 7.1. The map

Hom(K,G) −→ TeG, γ 7→ γ(0)

from the set Hom(K,G) of all one parameter subgroups of G to the tangentspace TeG of G at e is a bijection. Indeed, any γ ∈ Hom(K,G) is an integralcurve of the vector field X ∈ g with Xe = γ(0).

Before we give the proof we discuss the example G = GLn(K).

Example 7.2. For the general linear groupGLn(K) with Lie algebra gln(K) =Kn,n the left invariant vector field taking the value A ∈ Kn,n at E is, accord-ing to Example 5.3 given by Ξ 7→ (ΞA) · ∆, hence the differential equationfor the corresponding one parameter subgroup is γ = γA with the solutionγ(t) = etA, where

etA :=∞∑ν=0

tnAn

n!.

34

Proof. For γ ∈ Hom(K,G) and s ∈ K we have γ(s+ t) = γ(s)γ(t), hence

γ(s) = (λγ(s))∗(γ(0)) = (λγ(s))∗(Xe) = Xγ(s),

i.e. γ is an integral curve of X. Furthermore the injectivity is now alsoobvious, since an integral curve γ of a vector field X is completely determinedby its initial value γ(0). To get surjectivity, let us first consider the real case.Given Xe ∈ TeG, we know that there is an integral curve γ : I −→ G ofthe vector field X ∈ g, where I ⊂ R is some open interval containing 0.Let I ⊂ R be a maximal open interval such that γ can be defined on I.If a ∈ ∂I, we can easily extend γ by defining γ(t) as γ(a

2)γ(t − a

2) near a.

So necessarily ∂I = ∅ resp. I = R. Now argue as above in order to seethat a globally defined integral curve of a left invariant vector field is a oneparameter subgroup. In the complex case, we consider Ze ∈ TeG as a realtangent vector and denote γϑ : R −→ G the one parameter subgroup withγϑ = eiϑZe. Then reiϑ 7→ γϑ(r) defines an integral ”curve” γ : C −→ G ofZ ∈ g.

Corollary 7.3. The connected real Lie groups of dimension one are R andS1, the connected complex Lie groups of dimension one are C,C∗ and the toriC/Λ with a lattice Λ = Zω1 +Zω2 (where ω1, ω2 ∈ C are linearly independentover R.)

Proof. Take any nonconstant one parameter subgroup γ : K −→ G. SinceG is connected and γ(K) contains a neighborhood of e ∈ G we obtain G =γ(K). The kernel ker γ ⊂ K contains 0 as isolated point, hence is a discretesubgroup and G ∼= K/ ker(γ).

For R such subgroups are either trivial or of the form Zω with someω ∈ R \ 0, hence G ∼= R or G ∼= R/Zω ∼= S1. For C there is a furtherpossibility for ker γ, namely ker γ = Λ, a lattice. So either G ∼= C or G ∼= C∗or G ∼= C/Λ.

Note that in the real case R ∼= R>0 via the exponential function, whileC∗ ∼= R>0 × S1 resp. C/Λ ∼= S1 × S1 as real Lie groups. On the otherhand C/Λ ∼= C/Λ as complex Lie groups (or even as complex manifolds) iffΛ = αΛ with a nonzero complex number α ∈ C∗. In particular the complextori constitute a continuous family of pairwise non isomorphic Lie groups orcomplex manifolds!

35

In the next step we collect all one parameter subgroups in a family: For aleft invariant vector field X ∈ g denote γX ∈ Hom(K,G) the one parametersubgroup with γX(0) = Xe.

Definition 7.4. The exponential map

exp := expG : TeG −→ G

is defined as

exp(Xe) := γX(1),

with the left invariant vector field X ∈ g taking the value Xe ∈ TeG at e ∈ G.

Example 7.5. Let G = GLn(K). For A ∈ Kn,n ∼= gln(K) we have γA(t) =etA and thus exp(A) = eA.

Since the solution of the equation for the integral curve of a vector fielddoes not only depend differentiably on the prescribed initial value, but evenvaries differentiably with the vector field itself, we obtain that exp : TeG −→G is differentiable. We compute its differential at the origin:

Proposition 7.6. The Jacobian of the exponential map

T0(exp) : TeG ∼= T0((TeG)) −→ TeG

at the origin is the identity on TeG. In particular it induces a diffeomorphismexp |U : U −→ V from an open neighbourhood of 0 ∈ TeG onto an openneighbourhood V of e ∈ G.

Proof. The curve t 7→ tXe has tangent vectorXe at the origin, thus T0(exp)(Xe)is the tangent vector of t 7→ exp(tXe) = γtX(1) = γX(t). But γX(0) = Xe bydefinition.

In order to avoid cumbersome notation, we will from now on use capitalletters X, Y in order to denote both left invariant vector fields in g ⊂ Θ(G)as well as tangent vectors at e ∈ G, associated one to the other via the

evaluation isomorphism g∼=−→ TeG. With that convention we have

exp((s+ t)X) = γX(s+ t) = γX(s)γX(t) = exp(sX) exp(tX)

36

and thus exp(X + Y ) = exp(X) exp(Y ) if X and Y are linearly dependent.Otherwise, as a consequence of the fact that exp is a diffeomorphism near0 ∈ TeG we may write

exp(X) exp(Y ) = exp(C(X, Y ))

with a differentiable (holomorphic) function C : U × U −→ g and an openneighbourhood U ⊂ TeG ∼= g of 0. The interesting fact about the functionC : U × U −→ g is that it can be written as a convergent series

C(X, Y ) =∞∑n=1

Cn(X, Y ),

whereCn : g× g −→ g

is a homogeneous Lie bracket polynomial of degree n with rational coefficientsindependent of the Lie group G and the Lie algebra g. That is, Cn(X, Y ) isa rational linear combination of terms

[[...[Z1, Z2]..., Zn−1], Zn], Zi ∈ X, Y

with coefficients only depending on the function 1, ..., n −→ X, Y , i 7→ Zi(and not on the Lie algebra g.)

Thus the Lie algebra determines completely the group law in a neighbour-hood of e ∈ G: Using the local inverse log := (exp |U)−1 as a local coordinate,it is given by C : U × U −→ g. In order to obtain that result one derives adifferential equation for the g-valued function

F (t) := C(tX, tY )

defined on a neighbourhood of 0 ∈ K. For Z ∈ g denote ad(Z) : g −→ g thelinear map X 7→ [Z,X] (in fact a derivation as a consequence of the Jacobiidentity). Then

F (t) = f(ad(F (t)))(X + Y ) +1

2[X − Y, F (t)],

where f is a convergent power series in t with rational coefficients, indeed

f(t) =t

1− e−t− t

2.

37

Together with the initial condition F (0) = 0 ∈ g we obtain then a recursiveformula for the homogeneous Lie bracket polynomials Cn(X, Y ). We mentionhere only

C1(X, Y ) = X + Y, C2(X.Y ) =1

2[X, Y ]

and

C3(X, Y ) =1

12([[X, Y ], Y ]− [[X, Y ], X])

as well as

C4(X, Y ) = − 1

48([Y, [X, [X, Y ]]] + [X, [Y, [X, Y ]]]).

8 Subgroups and Subalgebras

In analogy to one parameter subgroups we define connected subgroups of aLie group G:

Definition 8.1. A connected (K-Lie) subgroup of a K-Lie group G is a pair(H, ι) with a connected Lie group H together with an injective (K-Lie group)homomorphism ι : H −→ G. Two such subgroups (H, ι), (H, ι) are equivalentif there is a Lie group isomorphism ψ : H −→ H with ι = ι ψ.

We remark that two connected subgroups are equivalent iff their imagegroups coincide. The image ι(H) is closed in G iff it is a submanifold (andthen H ∼= ι(H) even as Lie groups). The implication ”=⇒” is a nontrivialstatement we can not prove here. On the other hand, a submanifold ι(H)is locally closed and thus open in its closure ι(H), a connected topologicalgroup. But then ι(H) is an open subgroup of the connected topological groupι(H) and hence coincides with it.

Since ι∗ : h −→ g is a Lie algebra homomorphism we can associate toany Lie subgroup a Lie subalgebra of g := Lie(G), namely ι∗(h) ∼= h. Wewant to prove that every subalgebra h ⊂ g arises in that way. In order to getthe corresponding Lie subgroup one could look at exp(h). But that set neednot be even a group nor can we take its closure in G. Instead one has torealize the corresponding subgroup as a ”maximal integral submanifold of aninvolutive subbundle E ⊂ TG”. Let us start explaining these new notions:

Definition 8.2. A subbundle E ⊂ TM (of rank k) of the tangent bundleTM of a differentiable/complex manifold M is a union E :=

⋃x∈M Ex, where

38

Ex ⊂ TxM is a k-dimensional vector subspace of TxM for every x ∈M , suchthat every point a ∈ M has an open neighbourhood U ⊂ M together withvector fields X1, ..., Xk ∈ Θ(U), such that the tangent vectors X1,x, ..., Xk,x ∈TxM constitute a basis of Ex for all x ∈ U . We denote

ΘE(M) := X ∈ Θ(M);Xa ∈ Ea ∀ a ∈M

the vector subspace of all vector fields taking values in E ⊂ TM .

Note that any vector field X ∈ Θ(M) without zeros defines a subbundleE ⊂ TM , namely Ea := KXa for a ∈ M . Integral curves then generalize tointegral submanifolds:

Definition 8.3. An integral manifold of a subbundle E ⊂ TM is an in-jective immersion ι : N −→ M from a connected differentiable (complex)manifold N into M , such that Taι(TaN) = Ea for all a ∈ M . Two integralsubmanifolds ι : N −→ M and ι : N −→ M are called equivalent if there isa diffeomorphism (biholomorphic map) ψ : N −→ N with ι = ι ψ. We saythat two integral submanifolds agree at a ∈ M if a = ι(c) = ι(c) and ι|U isequivalent to ι|U for open neighbourhoods U, U of c, c.

Note that in the above situation every point c ∈ N admits an openneighbourhood U 3 c, such that ι(U) ⊂ M is a submanifold of M andι|U : U −→ ι(U) a diffeomorphism.

Example 8.4. Let M = Kn and E ⊂ T (Kn) be the subbundle spannedat each point by the values of the first k coordinate vector fields ∂1, ..., ∂k ∈Θ(Kn). Then, up to equivalence, the integral submanifolds of E are themaps

U −→ Kn, (x1, ..., xk) 7→ (x1, ..., xk, ak+1, ..., an)

with a connected open subset U ⊂ Kk.

Definition 8.5. A subbundle E ⊂ TM is called involutive (or an involutivesystem) if for all open subsets U ⊂M we have

X, Y ∈ ΘE(U) =⇒ [X, Y ] ∈ ΘE(U).

Example 8.6. 1. Take X = ∂1, Y = cos(x)∂2 + sin(x)∂3 ∈ Θ(K3) andEa := KXa +KYa. Then E =

⋃a∈K3 Ea ⊂ T (K3) is a subbundle, but

not an involutive one, since [X, Y ] = − sin(x)∂2 + cos(x)∂3 6∈ ΘE(K3).

39

2. A subbundle admitting at every point a ∈ M an integral submanifoldιa : Na −→M , i.e. with a ∈ ιa(Na), is involutive.

Theorem 8.7. Let E ⊂ TM be an involutive subbundle. Then every pointa ∈ M admits an open neighbourhood with local coordinates x1, ..., xn, suchthat, with the corresponding coordinate vector fields ∂1, ..., ∂n ∈ Θ(U), onehas Eb = K∂1,b + ...+K∂k,b for all b ∈ U .

Corollary 8.8. An involutive subbundle E ⊂ TM admits integral manifoldsat every point a ∈M , and any two such integral submanifolds agree near a.

Proof. We do induction on the rank k of the subbundle E ⊂ TM .

The case k = 1: Take a submanifold S 3 a of M of dimension n − 1 withXa 6∈ TaS, where X ∈ ΘE(U). Then the inverse of the flow

µ : S × I −→M

of the vector field X ∈ Θ(U) (here I ⊂ K denotes a small open interval ordisc containing 0 ∈ K) provides local coordinates with X = ∂1. Here theflow means the map

µ(x, t) := γx(t),

where γx : I −→M denotes the integral curve of X with initial value γx(0) =x.

The case k > 1: Denote X1, ..., Xk ∈ ΘE(U) spanning vector fields on aneighbourhood U of a ∈ M . From the case k = 1 we know that thereare local coordinates y1, ...., yn near a ∈ M , such that X1 = ∂y1 . We shallnow replace the remaining spanning vector fields X2, ..., Xk ∈ ΘE(U) withvector fields Y2, ..., Yk ∈ ΘE(U) tangent to any submanifold t ×Kn−1 andsatisfying

[X1, Yi] = 0 , i = 2, ...., k.

Indeed in that case we have

Yi =n∑j=2

fij∂yj

with coefficient functions

fij = fij(y2, ..., yn)

40

not depending on y1. Denote E0 ⊂ TV , where V ⊂ Kn−1 is an open neigh-bourhood of 0 ∈ Kn−1, the subbundle generated by the vector fields Yi.Indeed, it is involutive, since E is. Now E0 having rank k − 1 we find coor-dinates x2, ...., xn on V (may be after a shrinking) such that ∂2, ...., ∂k spanE0. Finally y1, x2, ..., xn are the local coordinates on Kn = K ×Kn−1 we arelooking for.The construction of the vector fields Y2, ..., Yk ∈ ΘE(U): We may assumethat the vector fields X2, ..., Xk ∈ ΘE(U) are tangent to t ×Kn−1 for allt ∈ K near the origin, i.e. that

Xj =n∑`=2

hj`∂y`

holds for j ≥ 2: Replace Xj with Xj−hj1X1 = Xj−hj1∂y1 . Now we considera vector field

Y =k∑j=2

ϕjXj ∈ ΘE(U),

and study how to choose the coefficient functions ϕ2, ..., ϕk in order to have

0 = [X1, Y ] =k∑j=2

∂y1 (ϕj)Xj + ϕj[X1, Xj].

Since E is involutive, we have

[X1, Xj] =n∑`=2

∂y1 (hj`) · ∂y` ∈ ΘE(U)

and thus

[X1, Xj] =k∑`=2

gj` ·X`,

where the vector field X1 does not show up in the expansion of [X1, Xj] ∈ΘE(U) as a linear combination of X1, ..., Xk. Hence

0 =k∑j=2

(∂y1 (ϕj)Xj + ϕj

k∑`=2

gj` ·X`

).

41

or equivalently

∂y1 (ϕj) =k∑`=2

g`jϕ`, j = 2, ..., k.

This is an ODE with respect to the variable y1, keeping fixed y2, ....., yn.Thus we may prescribe the functions ϕj(0, y2, ..., yn), j = 2, ..., k, and obtainuniquely determined functions y 7→ ϕj(y, y2, ..., yn) satisfying the above sys-tem of differential equations. Now taking ϕj(0, y2, ..., yn) ≡ δij for i = 2, ..., k,we obtain the vector fields Yi, i = 2, ..., k.

As a consequence we can easily compare two integral submanifolds ι :N −→ M and ι : N −→ M of an involutive subbundle E ⊂ TM , namely:There are open subsets U ⊂ N, U ⊂ N with ι(U) = ι(N) ∩ ι(N) = ι(U),such that ι|U and ι|U are equivalent. In particular we can apply Zorn’s lemmaand see that through every point a ∈ M there is a unique maximal integralsubmanifold (i.e. unique up to equivalence).

Proposition 8.9. Let G be a Lie group. The map

(H, ι) 7→ ι∗(h) ⊂ g

defines a bijection between the set of connected Lie subgroups of G (up toequivalence) and the Lie subalgebras of g = Lie(G).

Proof. Given a subalgebra h ⊂ g we define an involutive subbundle E ⊂ TGby Ea := Teλa(he) (with he := Xe;X ∈ h). Indeed, if X1, ..., Xk ∈ his a basis, then Ea = KX1,a ⊕ ... ⊕ KXk,a. In particular h ⊂ g being asubalgebra, we see that E ⊂ TM is involutive. Now take ι : H −→ G asa maximal integral submanifold of E at e ∈ G. To simplify notation wetreat ι as an inclusion. We then have aH = H for all a ∈ H: Since E isλa-invariant, aH 3 a is a maximal integral submanifold, but H 3 a is as well,hence aH = H. Since on the other hand e ∈ H, this immediately gives thatH ⊂ G is a subgroup (in the algebraic sense). We leave it to the reader tocheck, that H becomes a Lie group with this group law (note once again thatin general the topology and differentiable structure is only locally that oneinduced by G). On the other hand every connected K-Lie subgroup (H, ι)clearly is an integral submanifold of the left invariant (involutive) subbundleE ⊂ TM with Ee = Teι(TeH). It follows the injectivity of our map.

42

Let us now investigate what it means that ι(H) ⊂ G is a normal subgroupof G. In order to obtain a differential interpretation of that property weconsider the adjoint representation:

Definition 8.10. A representation of a (K-)Lie group G is a Lie grouphomomorphism G −→ GL(V ) from G into the general linear group GL(V )of a finite dimensional K-vector space V . The adjoint representation of aLie group G with Lie algebra g := Lie(G) is the homomorphism Ad : G −→GL(g), a 7→ Te(κa), where κa : G −→ G, g 7→ aga−1 is conjugation with a.

Proposition 8.11. The differential

ad := TeAd : g = Lie(G) −→ gl(g) = Lie(GL(g))

of the adjoint representation

Ad : G −→ GL(g), a 7→ Te(κa)

satisfiesad(X) = [X, ..]

Proof. We start with the equality

Ad(exp(X)) = ead(X)

of endomorphisms of g. Evaluating it at some Y ∈ g gives

Ad(exp(X))(Y ) = ead(X)Y

and then apply both sides to a function f ∈ C∞(G)

Y (f κexp(X)) = (Teκexp(X)(Y ))f = (ead(X)Y )f,

i.e.,Y f(exp(X) · .. · exp(−X)) = (ead(X)Y )f.

Now replace X with sX, s ∈ K, in order to get

Y f(exp(sX) · .. · exp(−sX)) = (es·ad(X)Y )f.

Differentiation at s = 0 yields:

d

ds(Y f(exp(sX) · .. · exp(−sX)))s=0 = (ad(X)Y )f

43

resp.

∂2

∂s∂t(f(exp(sX) exp(tY ) exp(−sX)))s=0,t=0 = (ad(X)Y )f.

The right hand side can be rewritten with the chain rule as

(∂2

∂s∂tf(exp(sX) exp(tY )) +

∂2

∂t∂sf(exp(tY ) exp(−sX)))s=t=0.

Since for a left invariant vector field Y ∈ g one has

d

dtf(a exp(tY ))t=0 = Y (f λa)(e) = ((Y f) λa)(e) = (Y f)(a)

we obtain (with a = exp(sX) resp. a = exp(tY )) finally

(XY f)(e) + (Y (−X)f)(e) = (XY f)(e)− (Y Xf)(e) = ([X, Y ]f)(e)

resp. [X, Y ] = ad(X)(Y ).

In order to simplify notation we shall from now on identify a connected Liesubgroup (H, ι) with its image ι(H) and its Lie algebra h with the subalgebraι∗(h) ⊂ g.

Corollary 8.12. A connected Lie subgroup H ⊂ G of a connected Lie groupG is a normal subgroup if and only if its Lie algebra h ⊂ g is an ideal, i.e.X ∈ g, Y ∈ h =⇒ [X, Y ] ∈ h.

Proof. The subgroup H is normal iff κa(H) = H for all a ∈ G. Obviouslythat implies Teκa(h) = h for all a ∈ G, in particular for a = exp(X) withX ∈ g. So h 3 Te(κexp(X))(Y ) = Ad(exp(X))(Y ) = ead(X)Y gives once again,after replacing X with sX, s ∈ K and differentiation with respect to s ats = 0 that

h 3 d

ds(es·ad(X)Y )s=0 = ad(X)(Y ) = [X, Y ],

the subalgebra h ⊂ g being closed in g. (Every finite dimensional K-vectorspace carries a natural topology obtained from a linear isomorphism V ∼= Kn

and not depending on the choice of that isomorphism.)On the other hand let us assume that the subalgebra h ⊂ g is even an

ideal. We show that the normalizer

NG(H) := a ∈ G;κa(H) = H

44

contains an open neighbourhood U ⊂ G of the neutral element e ∈ G. Thegroup G being connected, it follows NG(H) = G. Since exp(g) containssuch a neighbourhood, it suffices to consider a = exp(X) with X ∈ g. Butad(X)(Y ) ∈ h for Y ∈ h implies

Teκexp(X)Y = Ad(exp(X))(Y ) = ead(X)Y ∈ h.

Finally if we can prove that κexp(X) maps an open neighbourhood of e ∈ Hinto H, we are done, since H as connected group is generated by any openneighbourhood of e. But any element in such a neighbourhood can again betaken in the form exp(Y ) and then

κexp(X)(exp(Y )) = exp(Teκexp(X)Y ) ∈ exp(h) ⊂ H.

9 Lie Algebras of Dimension ≤ 3

For Lie algebras there is as well the notion of a derivation:

Definition 9.1. Let g be a Lie algebra. A linear map D : g −→ g is calleda derivation, if it satisfies the Leibniz rule with respect to the Lie bracket:

D[X, Y ] = [DX, Y ] + [X,DY ], ∀ X, Y ∈ g.

We denote Der(g) ⊂ gl(g) the subalgebra of gl(g) consisting of all derivationsof g.

Remark 9.2. 1. We have idg ∈ Der(g) iff g is abelian. Indeed in thatcase Der(g) = gl(g).

2. Given any element Z ∈ g, the map ad(Z) : g −→ g, X 7→ [Z,X] is aderivation: In fact the Leibniz rule is nothing but the Jacobi identityfor Z,X, Y .

3. The Jacobi identity may even be reformulated in a more sophisticatedway by saying that the map g −→ gl(g), Z 7→ ad(Z), is a homomor-phism of Lie algebras, the adjoint representation of g. (A representationof a Lie algebra is any Lie algebra homomorphism g −→ gl(V ), whereV is a K-vector space.)

45

4. If λ, µ ∈ K are eigenvalues of a derivation D ∈ Der(g), then eitherso is λ + µ or [X, Y ] = 0 for any two eigenvectors X, Y ∈ g of Dcorresponding to λ resp. µ.

5. If D ∈ Der(g) and dim g <∞, then eD : g −→ g is a Lie algebra auto-morphism, i.e., a linear isomorphism with eD([X, Y ]) = [eDX, eDY ] forall X, Y ∈ g.

6. If D ∈ Der(h), we can form a new Lie algebra hD. As vector space itis the cartesian product (or direct sum)

hD := h×K,

where Z := (0, 1) ∈ hD satisfies [(0, 1), (X, 0)] = DX. Here, the factthat D is a derivation gives the Jacobi identity on hD. Note thathD ∼= hD, if D is similar to λD for some λ ∈ K∗. Furthermore

[hD, hD] = ([h, h] +Dh)× 0.

Remark 9.3. If H is a Lie group with Lie algebra h, then a Lie group withLie algebra hD should be a semidirect product, e.g.

H ×σ K,

where σ : K −→ Aut(H) is a homomorphism, i.e. σt+s = σt σs. Here weneed that

σt : H −→ H

satisfiesTe(σt) = etD : h −→ h, ∀ t ∈ K.

In general it is not true, that the automorphism etD is induced by some(indeed unique, if it exists) σt : H −→ H. But such a ”lifting” exists always,if H is simply connected. For example let us consider h = K,D = id. ForK = R we could take H = R>0 ⊂ R∗. Then σt(x) = xt is the lifting weare looking for. But if H = C∗ and D = idC : C −→ C, there is someσt : C∗ −→ C∗ only for t ∈ Z.

Om the other hand, if we remember that the Lie algebra of K∗ is K aswell, we may try

H ×σ K∗

46

and succeed with (now we need σts = σt σs)

σt : K −→ K, x 7→ tx.

The group we obtain is isomorphic to

(K∗ K0 1

), or as well to the group

of affine linear transformations of K.

For two subspaces U, V ⊂ g we denote

[U, V ] := span[X, Y ];X ∈ U, Y ∈ V .

Now the commutator [g, g] of a Lie algebra g is not only a subalgebra, buteven an ideal, and any isomorphism between two Lie algebras transformsthe respective commutators one to the other: It is a ”characteristic ideal”.Indeed if X1, ..., Xn is a basis of the K-vector space g, then [g, g] is spannedby the Lie brackets [Xi, Xj], i < j.

Let us now come to our classification:

1. dim g = 1: A one dimensional Lie algebra is obviously abelian.

2. dim g = 2: Any two dimensional non-abelian Lie algebra g is of the formg = hid with a one dimensional (abelian) Lie algebra h: Necessarilydim[g, g] = 1, say [g, g] = KZ. If we take any X 6∈ [g, g], then [X,Z] =λZ with λ ∈ K∗, and after a rescaling of X we may assume [X,Z] = Z.We need later on the following

Lemma 9.4. For the two dimensional non-abelian Lie algebra g onehas

Der(g) = D ∈ gl(g);Dg ⊂ [g, g].

Proof. It is easily checked, that a linear map D : g −→ g satisfyingDX = λZ,DZ = µZ is a derivation.

Now let D : g −→ g be a derivation. Since D[g, g] ⊂ [g, g], we haveDZ = λZ. For λ = 0 we get

0 = D[X,Z] = [DX,Z] + [X,DZ] = [DX,Z]

and thus DX ∈ KZ = [g, g] as desired. Now assume λ 6= 0. If D isdiagonalizable we may assume that DX = µX with a µ ∈ K. Since

47

[X,Z] = Z, we conclude µ + λ = λ, i.e. µ = 0 and thus Dg ⊂ KZ =[g, g]. If D is not diagonalizable, we have DX − λX ∈ K∗Z. But thenZ = [X,Z] satisfies

D[X,Z] = [DX,Z] + [X,DZ] = [λX + βZ,Z] + [X,λZ] = 2λ[X,Z].

So λ = 2λ, i.e. λ = 0, a contradiction.

3. dim g = 3:

(a) dim[g, g] = 0: Then g is abelian.

(b) dim[g, g] = 1, say [g, g] = KZ:

i. If [Z, g] = 0 (or equivalently [[g, g], g] = 0), we have

g ∼= (K2)D,

where K2 is the two dimensional abelian Lie algebra and D :K2 −→ K2 a nontrivial nilpotent derivation (linear map).Indeed, we find a basis X, Y, Z with [X, Y ] = Z. Then a :=KX + KZ is an abelian ideal and we find g = aD with thenilpotent derivation D : a −→ a, X 7→ −Z 7→ 0.

ii. Otherwise [Z, g] = KZ (or equivalently [[g, g], g] = [g, g]).Then we have

g ∼= h⊕K (∼= h0),

with the two dimensional non-abelian Lie algebra h. Wechoose X ∈ g with [X,Z] = Z. Choose then Y 6∈ KX + KZwith [Y, Z] = 0. That is possible, since [X, g] = KZ in anycase. Furthermore we can assume [X, Y ] = 0; if that was notthe case right from the beginning, we may replace Y withY −λZ, where [X, Y ] = λZ with λ ∈ K∗. Hence g = h⊕KYis the direct sum of the two dimensional non-abelian Lie alge-bra h and a one dimensional Lie algebra. We could also writeg ∼= h0, where 0 ∈ Der(h) denotes the trivial derivation.

(c) dim[g, g] = 2: Then we have

g = (K2)D,

where K2 denotes the two dimensional abelian Lie algebra andD ∈ GL2(K) is any automorphism of the vector space K2. To see

48

that, take any X 6∈ a := [g, g] and D := ad(X)|a. By Lemma 9.4,if a was not abelian, we had Da ⊂ [a, a] and hence

a = [g, g] = [a, a] +Da = [a, a],

a contradiction to our assumption dim a = 2. So a is abelian andDan isomorphism because of a = [g, g] = Da. So g = aD with a twodimensional abelian Lie algebra a and any linear automorphismD ∈ gl(a) = Der(a).

4. dim[g, g] = 3: Given a basis X1, X2, X3 of g, the Lie brackets [X1, X2],[X1, X3], [X2, X3] form a basis as well because of [g, g] = g.

First we hunt for some element X ∈ g \ 0, such that the linear map

ad(X) = [X, ..] : g −→ g

is easily understood. Since [g, g] = g, its image [X, g] is a two di-mensional subspace and ker(ad(X)) = KX. We now are looking foran ad(X)-invariant subspace, complementary to KX. Let us try with[X, g] = ad(X)(g): Indeed if X 6∈ [X, g], we have

g = KX ⊕ [X, g]

as vector spaces. So we first show that the assumption

∀ X ∈ g : X ∈ [X, g]

leads to a contradiction: Given any X ∈ g \ 0, choose Y ∈ g withX = [X, Y ] and then Z ∈ g with Y = [Y, Z]. We claim that theelements X, Y, Z are linearly independent. Apply D := ad(Y ) to arelation

αX + βY + γZ = 0

and obtain−αX + γY = 0.

Since X, Y are linearly independent, that implies α = 0 = γ, whenceβ = 0 as well. Hence we can write

[X,Z] = αX + βY + γZ

49

and thusD[X,Z] = α[Y,X] + γ[Y, Z],

while on the other hand

D[X,Z] = [DX,Z] + [X,DZ] = −[X,Z] + [X, Y ].

Since the elements [X, Y ], [Y, Z], [X,Z] are linearly independent, thatis not possible.

Now let us return to the map D := ad(X), where X 6∈ [X, g]. It inducesa linear automorphism of [X, g]. If it is diagonalizable there are twolinearly independent eigenvectors Y, Z ∈ [X, g], say DY = λY,DZ =µZ. Since [Y, Z] 6= 0, we find λ + µ ∈ λ, µ, 0 – the eigenvalues ofad(X) – resp. λ + µ = 0 because of λ, µ ∈ K∗. After a rescaling of Xwe can assume λ = 2, µ = −2. Finally 0 6= [Y, Z] ∈ ker(ad(X)) = KX,such that we after a rescaling of Y , say, obtain [Y, Z] = X. Indeed, thelinear map

g∼=−→ sl2(K)

with

X 7→(

1 00 −1

), Y 7→

(0 10 0

), Z 7→

(0 01 0

)turns out to be an isomorphism of Lie algebras.

Now let us exclude the possibility that D is not diagonalizable withone eigenvalue λ 6= 0. Then we find an eigenvector Z ∈ [X, g], i.e.DZ = λZ, and a vector Y ∈ [X, g] with DY − λY = Z, and thus

D[Y, Z] = [DY,Z] + [Y,DZ] = [λY + Z,Z] + [Y, λZ] = 2λ[Y, Z].

Since [Y, Z] 6= 0 that implies 2λ ∈ 0, λ, but that is impossible.

This finishes our discussion for K = C. For K = R we have to dealwith the case that D = ad(X)|[X,g] has no real eigenvalus. We pass tothe complexification gC = g⊕ ig. The complexification DC := D + iDhas two eigenvalues λ,−λ. Since they are zeros of a real polynomial,we have −λ = λ, i.e. λ is purely imaginary. After a (real) rescalingof X we may thus assume λ = i and thus D2 = −id[X,g]. Now forany Y ∈ [X, g] the elements Y, Z := DY form a basis of [X, g], andobviously DZ = −Y . Hence D[Y, Z] = [DY,Z] + [Y,DZ] = 0 and thusker(D) = KX tells us that [Y, Z] = λX with a λ ∈ R∗. By dividing

50

Y, Z with√|λ| we finally get [Y, Z] = ±X. If [Y, Z] = X we obtain the

vector product multiplication table of the vectors in an orthonormalbasis of R3. More explicitly there is an isomorphism

g∼=−→ so3(R)

with

X 7→

0 0 00 0 10 −1 0

, Y 7→

0 0 10 0 0−1 0 0

, Z 7→

0 1 0−1 0 00 0 0

.

Using complex matrices we have even an isomorphism

g∼=−→ su2 ⊂ sl2(C)

with

X 7→(i 00 −i

), Y 7→

(0 1−1 0

), Z 7→

(0 ii 0

).

Note that su2∼= so3(R) is not isomorphic to sl2(R) since in the former

algebra no derivation ad(X), X ∈ g \ 0 is diagonalizable.

If on the other hand [Y, Z] = −X the derivation ad(Y ) is diagonalizablewith the eigenvalues 1,−1 and corresponding eigenvectors X−Z,X+Z.So g ∼= sl2(R) in that case.

10 The Universal Covering Group

A Lie group homomorphism ϕ : G −→ H induces a Lie algebra homomor-phism ϕ∗ : g −→ h. In this section we ask when for given Lie groups G,Hand a Lie algebra homomorphism ψ : g −→ h we can find a Lie group homo-morphism ϕ : G −→ H inducing ψ, i.e. such that ψ = ϕ∗. The strategy isas follows: If ϕ : G −→ H is a Lie group homomorphism, then its graph

Γϕ := (g, ϕ(g)); g ∈ G ⊂ G×H

is a Lie subgroup of G×H with Lie algebra

Lie(Γϕ) = (X,ϕ∗(X));X ∈ g.

51

So given ψ : g −→ h we look at the connected Lie subgroup Γ ⊂ G×H with

Lie(Γ) = (X,ψ(X));X ∈ g.

The inclusion followed by the projection onto the first factor

π : Γ → G×H prG−→ G

has obviously bijective differential

π∗ : Lie(Γ) −→ g.

Since both groups, G and Γ are connected, it is a surjective homomorphismwith discrete kernel. And if it is even an isomorphism, we can take ϕ :=prH π−1. Indeed, there are groups, where π necessarily is an isomorphism.

In order to understand that phenomenon we need an excursion to topol-ogy. The basic notion is that of of a covering :

Definition 10.1. A continuous map π : X −→ Y between topological spacesX and Y is called a covering iff every point b ∈ Y admits an open neigh-bourhood V ⊂ Y , such that its inverse image is the disjoint union

π−1(V ) =⋃i∈I

Ui

of open subsets Ui ⊂ X with π|Ui: Ui −→ V being a homeomorphism for

every i ∈ I.

Example 10.2. A surjective Lie group homomorphism ϕ : G −→ H with aconnected Lie group G and discrete kernel D ⊂ G (e.g. if ϕ∗ : g −→ h is anisomorphism) is a covering: Choose an open neighbourhood U ⊂ G of e ∈ Gwith U · U−1 ⊂ G \D∗, where D∗ := D \ e. Then for V := π(U) we have

π−1(V ) =⋃a∈D

aU,

a disjoint union.

Now, given a covering π : X −→ Y we want to study spaces Z, such thatany continuous map ϕ : Z −→ Y admits a lift ϕ : Z −→ X, i.e. such thatϕ = π ϕ. For Z = [0, 1], the unit interval in R, we have:

52

Proposition 10.3. Let π : X −→ Y be a covering of the locally path con-nected space Y . Then, given a path γ : [0, 1] −→ Y and a point x0 ∈X, π(x0) = γ(0), there is a unique path γ : [0, 1] −→ X with γ(0) = x0.

Proof. Write [0, 1] = I1 ∪ ... ∪ In with Ik = [k−1n, kn]. For sufficiently big

n ∈ N every piece γ(Ik) ⊂ γ(I) is contained in an open path connected setV = Vk ⊂ Y , such that π−1(V ) is a disjoint union as in Def. 10.1. Nowassume we have found a lift

γk :[0, k/n

]−→ X,

of γk := γ|[0, kn

]. Choose U ⊂ π−1(Vk+1) with π|U : U −→ Vk+1 being homeo-

morphic and γk(kn) ∈ U . Now define γk+1 by

γk+1|[0,k/n] := γk , γk+1|Ik+1:= (π|U)−1|Ik+1

.

The idea now is to fix a base point z0 in a (path connected) topologicalspace Z and to define for a given continuous map ϕ : Z −→ Y a lift asfollows: Choose a point x0 ∈ X above y0 := ϕ(z0) (i.e. π(x0) = y0). Now,given a point z ∈ Z, take a path βz : [0, 1] −→ Z with βz(0) = z0, βz(1) = z.Denote γz : [0, 1] −→ X the lift of γz := ϕ βz with γz(0) = x0 and takeϕ(z) := γz(1). It remains to show that under suitable assumptions on Zdifferent choices of βz : [0.1] −→ Z give the same value ϕ(z).

For that we need the notion of homotopic paths :

Definition 10.4. Two paths α, β : [0, 1] −→ Y with same start and endpoint are called homotopic: ”α ∼ β”, if there is a homotopy from α to β,i.e., a continuous map F : [0, 1]× [0, 1] −→ Y with the following properties:

F (0, t) = α(0) = β(0), F (1, t) = α(1) = β(1)

and Ft(s) := F (s, t) satisfies

F0 = α, F1 = β.

Remark 10.5. 1. To be homotopic is an equivalence relation on the setof paths from a given point x ∈ Y to another given point y ∈ Y . Wedenote [γ] the equivalence class (homotopy class) of the path γ. Ifτ : [0, 1] −→ [0, 1] is a continuous map with τ(0) = 0, τ(1) = 1 (a”reparametrization“), then γ τ ∼ γ.

53

2. Given paths α, β : [0, 1] −→ Y , such that β(0) = α(1), we define theconcatenation αβ : [0, 1] −→ Y by

(αβ)(s) =

α(2s) , if 0 ≤ s ≤ 1

2

β(2s− 1) , if 12≤ s ≤ 1

3. If α ∼ α, β ∼ β and the end point of α is the starting point of β, thenαβ ∼ αβ, in particular we can concatenate homotopy classes. Notethat in general α(βγ) 6= (αβ)γ, but that α(βγ) ∼ (αβ)γ, i.e. on thelevel of homotopy classes concatenation becomes associative.

4. We can not only compose paths, but there is also the notion of aninverse path: Given α : [0, 1] −→ Y , we denote α−1 : [0, 1] −→ Y thepath α−1(s) := α(1− s). Note that α−1α ∼ α(0) ∼ αα−1.

Proposition 10.6. Let π : X −→ Y be a covering and α, β : [0, 1] −→ Yhomotopic paths. Then a homotopy F : [0, 1]2 −→ Y between α and β can belifted to a homotopy F : [0, 1]2 −→ X between any two lifts α, β : [0, 1] −→ Xof α, β with the same starting point. In particular they have the same endpoint as well: α(1) = β(1).

Proof. Fix n ∈ N. We consider the subdivision of the unit square

[0, 1]2 =⋃

1≤i,j≤n

Qij

with

Qij :=

[i− 1

n,i

n

]×[j − 1

n,j

n

], 1 ≤ i, j ≤ n.

For sufficiently big n ∈ N every F (Qij) ⊂ Y is contained in an open connectedset V = Vij ⊂ Y , such that π−1(V ) is a disjoint union as in Def. 10.1. Nowassume we have found a lift

Fij : Bij :=⋃

(k,`)≺(i,j)

Qk` −→ X,

of F |Bij, where ≺ is the lexicographic order on 1, ..., n2. Since Bij ∩Qij is

connected, we have Fij(Bij∩Qij) ⊂ U for one of the subset U ⊂ π−1(V ) with

π|U : U −→ V being homeomorphic. Hence we may extend Fij to Bij ∪ Qij

defining it on Qij as (π|U)−1.

54

Definition 10.7. A path connected topological space Z is called simply con-nected if it is connected and any closed path γ : [0, 1] −→ Z is nullhomotopic,i.e. homotopic to the constant path ≡ γ(0) = γ(1).

Remark 10.8. 1. It suffices to check the above condition for one basepoint as prescribed starting and end point for the closed path γ.

2. In a simply connected topological space Z any two paths with the samestart and the same end points are homotopic.

Example 10.9. 1. Obviously Kn is simply connected.

2. A path connected space X = U ∪ V , which is the union of two opensimply connected subsets U, V ⊂ X with a path connected intersectionU ∩ V , is simply connected. In particular the spheres Sn, n ≥ 2, aresimply connected. – To see this take a base point x0 ∈ U ∩ V andconsider a closed path γ : [0, 1] −→ X. Then for sufficiently big n ∈ Nevery interval Ik := [k−1

n, kn] satisfies γ(Ik) ⊂ U or γ(Ik) ⊂ V . Choose

a path αk from x0 to γ( kn) within U resp. V if γ( k

n) ∈ U resp. γ( k

n) ∈

V . That is possible, since U ∩ V is connected. Then γ ∼ β1...βn :=(..(β1β2)....βn) with β1 := γ1α

−11 , βk := αk−1γkα

−1k , 2 ≤ k < n and βn :=

αn−1γn. Since both U and V are simply connected and βk([0, 1]) ⊂ Uor βk([0, 1]) ⊂ V , we get βk ∼ x0, 1 ≤ k ≤ n, and thus γ ∼ x0.

As a consequence of Proposition 10.6 we obtain:

Proposition 10.10. Let π : X −→ Y be a covering and ϕ : Z −→ Y be acontinuous map from the simply connected topological space Z to Y . Thengiven points z0 ∈ Z, x0 ∈ X with ϕ(z0) = π(x0), there is a unique liftingϕ : Z −→ X of ϕ with ϕ(z0) = x0.

Corollary 10.11. A covering π : X −→ Y from a connected space X to asimply connected space Y is a homeomorphism. In particular for a simplyconnected Lie group G the mapping

Hom(G,H) −→ Hom(g, h), ϕ 7→ ϕ∗

is bijective for any Lie group H.

Proof. The identity id := idY : Y −→ Y admits a lifting id. Its image in Xis non-empty and both open and closed, hence equals X.

55

Definition 10.12. A covering π : X −→ X is called the universal coveringof X, if X is simply connected.

We call a topological space locally simply connected if every point has anopen simply connected neighbourhood.

Theorem 10.13. Every connected and locally simply connected topologicalspace X admits a covering π : X −→ X with a simply connected X, calledthe universal covering of X.

Proof. The construction is motivated by the following observation: Assumethere is a universal covering π : X −→ X and choose base points x0 ∈ Xas well as x0 ∈ X above x0 ∈ X, i.e. π(x0) = x0. Then the points in thefiber π−1(x) of a point x ∈ X are in one-to-one correspondence with thehomotopy classes of paths in X from the base point x0 ∈ X to x: Given apoint x ∈ π−1(x), choose a path γ from x0 to x. Then associate to x thehomotopy class [π γ]; indeed that provides a well defined map, X beingsimply connected. On the other hand, given a homotopy class [γ] of a pathγ : [0, 1] −→ X from x0 to x, denote γ : [0, 1] −→ X the unique lift of γwith γ(0) = x0. The point x ∈ π−1(x), we associate to [γ] then is x := γ(1),”welldefinedness” being guaranteed by Proposition 10.6.

So in order to obtain the universal covering we may apply the followingstrategy: We choose a base point x0 ∈ X and define X to be the set of allhomotopy classes of paths with the base point x0 as start point. The mapπ : X −→ X then is defined as π([γ]) := γ(1). The topology on X is definedas follows: Given a point x := [γ] with x := π(x) and a simply connectedneighbourhood U of x, we set

U(x) := [γδ]; δ : [0, 1] −→ U, δ(0) = x , x = [γ].

Then the sets U(x) with an open neighbourhood U ⊂ X of x ∈ X constitutea basis for the topology of X. We claim, that for simply connected U wehave U(a) ∩ U(b) = ∅ for a, b ∈ π−1(x), a 6= b. Let a = [α], b = [β]. Assumethat [αδ] = [βδ′] with paths δ, δ′ : [0, 1] −→ U . Take a path γ : [0.1] −→ Ufrom δ(1) = δ′(1) to α(1) = β(1). Now, U being simply connected, we haveδγ ∼ 0 ∼ δ′γ and thus

αδ ∼ βδ′ =⇒ αδγ ∼ βδ′γ =⇒ α ∼ β,

a contradiction. It follows easily that X is Hausdorff and π : X −→ X acovering.

56

Finally we show that X is simply connected: Let α : [0, 1] −→ X bea closed path with start end point x0 := [x0] (where x0 is regarded as theconstant path). Consider the path α := π α. We have α(1) = [α], since αis a lift of α with starting point x0 as well as the path t 7→ [αt] with the pathαt : [0, 1] −→ X, s 7→ α(ts). So because of the unique lifting property weobtain α(1) = [α1] = [α]. But α was a closed path, i.e. [α] = α(1) = α(0) =x0, i.e. α ∼ x0. By Proposition 10.6 we obtain α ∼ x0.

Now let us consider the situation where X = G is the topological spaceunderlying a connected Lie group.

Proposition 10.14. The universal covering G of a connected Lie group Gcarries, after the choice of a neutral element e ∈ π−1(e), a unique groupstructure, such that π : G −→ G becomes a group homomorphism.

Proof. For every point a ∈ G we define the map λa : G −→ G as the uniquelift of the map λa π : G −→ G, a := π(a), satisfying λa(e) = a. We leave itto the reader to check that the resulting map

G× G −→ G, (a, x) 7→ λa(x)

defines a group law on G, such that π : G −→ G becomes a group homomor-phism.

Remark 10.15. Fundamental Group: The construction of the universalcovering π : X −→ X can be used to associate to any connected and locallysimply connected space a group, namely the set

Deck(X) := f : X −→ X homeomorphism;π f = π

of all π-fiber preserving homeomorphisms of X (”deck transformations”)with the composition of maps as group law. From our above reasoningit follows that, given a base point x0 ∈ X, the restriction Deck(X) −→S(π−1(x0)), f 7→ f |π−1(x0) is injective. On the other hand, given pointsa, b ∈ π−1(x0) there is exactly one f ∈ Deck(X) with f(a) = b. If one wantsto avoid the universal covering π : X −→ X in the definition of Deck(X),one can construct an isomorphic group as follows: Take again a base pointx0 ∈ X and define the fundamental group of X as the set

π1(X, x0) := [γ]; γ path in X, γ(0) = x0 = γ(1)

57

of homotopy classes of closed paths in X with start and end point x0, thegroup law being the concatenation of paths representing homotopy classes:

[α][β] := [αβ].

Then there is a natural isomorphism

π1(X, x0) ∼= Deck(X)

as follows: Given [γ] take any lifting γ of γ, then the unique f ∈ Deck(X)with f(γ(1)) = γ(0) is the image of [γ].

Furthermore note that a path α from x0 ∈ X to x1 ∈ X induces anisomorphism

π1(X.x0)∼=−→ π1(X, x1), [γ] 7→ [α−1γα].

If G is a connected Lie group, we have

π1(G, e) ∼= Deck(G) ∼= K := ker(π : G −→ G),

where we associate to a ∈ K the left translation λa : G −→ G. Now adiscrete normal subgroup of a Lie group G is central, i.e. contained in itscenter Z(G), and hence abelian. So we have seen:

Corollary 10.16. The fundamental group π1(G, e) of a connected Lie groupis abelian.

In order to compute the fundamental group of a connected Lie group G,one often considers a connected closed Lie subgroup H ⊂ G, usually realizedas the stabilizer H = Gx of a point x ∈ M , when given a differentiableG-action G × M −→ M on some manifold M . Then E := G −→ B :=G/H ∼= Gx is an H-principal bundle, and in that situation we can apply thefollowing theorem:

Theorem 10.17. Let H be a connected Lie group and p : E −→ B be an H-principal bundle over the path connected base B. Choose base points b0 ∈ Band x0 ∈ π−1(b0) ⊂ E. Then there is an exact sequence

π1(H, e)ι∗−→ π1(E, x0)

p∗−→ π1(B, b0) −→ 1,

where ι : H −→ E, h 7→ x0h identifies (H, e) with (p−1(b0), x0), and ι∗, p∗denote the induced homomorphisms on the level of the fundamental groups.Recall that exactness means:

ker(p∗) = im(ι∗), im(p∗) = π1(B, b0).

58

Proof. Since the map p : E −→ B is locally trivial (i.e. B can be covered

with open subsets U , such that U×H ∼= p−1(U)p−→ U ⊂ B is the projection

onto U), every path γ : I −→ B admits a lifting to a path γ : I −→ E. Thisimplies the surjectivity of p∗. Furthermore obviously p∗ι∗ ≡ [b0] ∈ π1(B, b0).

It remains to show the inclusion ker(p∗) ⊂ im(ι∗): So let γ : I −→ E bea closed path with start and end point x0, such that p γ is nullhomotopicin B. We have to find a homotopy between γ and a suitable closed path inp−1(b0). Consider a homotopy F : I2 −→ B with F0 = p γ, F1 ≡ b0. Asin the proof of Proposition 10.6 we are looking for a lifting F : I2 −→ E ofF with F0 = γ, Ft(0) = x0 = Ft(1) for all t ∈ I. Then F1 : I −→ E is thedesired path in the fiber over b0 homotopic to γ.

Subdivide again I2 into small squares Qij, such that Qij ⊂ Uij for someopen set Uij with p−1(Uij) ∼= Uij ×H. Define

Bij := B ∪⋃

(k,`)≺(i,j)

Qk`

with B := I × 0 ∪ ∂I × I. On B00 = B the lift is given by F0 = γ, Ft(0) =x0 = Ft(1). Now assume we have found a lift Fij : Bij −→ E of F |Bij

. Takean open subset U ⊃ Qij with p−1(U) ∼= U ×H. Denote

q : p−1(U) ∼= U ×H −→ H

the projection onto H. Now the intersection Bij ∩Qij is connected and neverthe entire boundary ∂Qij of the tiny square Qij, but consists of two or threeof its sides. So we can first run through the sides contained in Bij ∩Qij and

then through the remainig ones, defining there an extension of q Fij, such

that one simply returns to the starting point of q Fij|Bij∩Qijby inverting

the parametrization. The resulting closed path ∂I2 −→ H is nullhomotopicin H, hence it can be extended to a continuous map f : Qij −→ H. Now

extend Fij to Bij by defining it on the square Qij as

Qij −→ U ×H ∼= p−1(U), (s, t) 7→ (F (s, t), f(s, t)).

11 Structure Theory for Lie Algebras

The last section gives a brief survey without proofs over some importantresults on the structure of real and complex Lie algebras. We start with the

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following ”embedding theorem”:

Theorem 11.1 (Theorem of Ado). Any finite dimensional K-Lie algebra gis isomorphic to a subalgebra h ⊂ gln(K) for some n ∈ N.

As a consequence we see that

1. any Lie algebra g is isomorphic to the Lie algebra Lie(G) of a Lie groupG, and

2. any connected Lie group G is ”locally isomorphic” to a linear group H(:= a connected subgroup H ⊂ GLn(K) for some n ∈ N), i.e., theirsimply connected covering groups are isomorphic: G ∼= H. Indeed, thatis the best we can say, since a factor group G/K with a discrete normal(indeed central) subgroup K ⊂ G or the universal covering group G ofa linear group G need not be linear again!

In Remark 9.2.6 we have discussed a procedure how to construct a new Liealgebra gD starting from a Lie algebra g and a derivation D ∈ Der(g). Theclass of all solvable Lie algebras, see Def.11.3, turns out to consist exactly ofthose Lie algebras which can be obtained from the abelian Lie algebra g = Kby an iterated application of that step.

Remark 11.2. Let g = Lie(G) with a simply connected Lie group G andD ∈ Der(g). For t ∈ K denote σt : G −→ G the automorphism with (σt)∗ =etD : g −→ g. Then σ : K −→ Aut(G), t 7→ σt, is a group homomorphism,and the semidirect product G×σ K is the simply connected Lie group withLie algebra isomorphic to gD. Remember that the underlying manifold isjust the cartesian product G×K, while the group law looks as follows:

(g, t)(g′, t′) = (gσt(g′), t+ t′).

In particular we see that the manifold underlying a simply connected solvableLie group is isomorphic to Kn for some n ∈ N.

Back to Lie algebras! Note first that, given an ideal a ⊂ g of a Lie algebrag we can endow the factor vector space with a natural Lie bracket

[X + a, Y + a] := [X, Y ] + a,

the resulting Lie algebra being called the factor algebra g/a of g mod(ulo)the ideal a. Furthermore that g = aD if dim g = dim a + 1 and D = ad(X)for some X ∈ g \ a.

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Definition 11.3. The Lie algebra g is called solvable if there is a finitesequence of subalgebras

g0 = 0 ⊂ g1 ⊂ ... ⊂ gr = g,

such that gi ⊂ gi+1 is an ideal of gi+1 for i < r with abelian factor algebragi+1/gi (or equivalently [gi+1, gi+1] ⊂ gi).

Example 11.4. 1. The Lie algebra utn(K) := Lie(UTn(K)) is solvable.Indeed take

gi := A ∈ utn(K);A(Vk) ⊂ Vk−(n−i) k = 0, ..., n,

where we set Vk = Kk × 0 ⊂ Kn and Vk = 0 for k ≤ 0. ForA,B ∈ gn we have [A,B] ∈ gn−1, since gl(Vk/Vk−1) is abelian, while forA,B ∈ gi, i < n, we already have AB,BA ∈ gi−1.

2. If g is solvable, then any subspace h ⊂ g with gi ⊂ h ⊂ gi+1 is asubalgebra and even an ideal in gi+1. Hence we may refine a givenstrictly increasing sequence as in Def. 11.3 in such a way that finallyr = dim g and dim gi+1 = dim gi + 1. In particular we see, that asolvable algebra can be constructed by a repeated application of the gD-construction for a Lie algebra g together with a derivation D ∈ Der(g).

3. Denote C(g) := [g, g] the commutator subalgebra of g. A Lie alge-bra is solvable iff the decreasing sequence of successive commutatorsubalgebras Ci(g), i.e.

C0(g) := g, Ci+1(g) := C(Ci(g))

terminates at the trivial subalgebra.

4. Let a ⊂ g be an ideal. If a is solvable as well as g/a, so is g. Inparticular, if a, b ⊂ g are solvable ideals, so is a + b. Hence there is aunique maximal solvable ideal in a Lie algebra g:

Definition 11.5. The maximal solvable ideal r ⊂ g is called the radical ofthe Lie algebra g.

Example 11.6. For g = gln(K) we find r = KE.

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Obviously the factor algebra g/r has trivial radical. The next theoremtells us that there is a subalgebra h ⊂ g projecting isomorphically onto thefactor algebra g/r. It is clear that there is a subspace U ⊂ g, such thatg = r ⊕ U and thus U −→ g −→ g/r is an isomorphism of vector spaces.The point is, that we can even find a complementary subspace, which isclosed with respect to the Lie bracket, with other words a complementaryLie subalgebra.

Theorem 11.7. For any Lie algebra g there is a subalgebra h (a ”Levi sub-algebra”) complementary to the radical r ⊂ g, i.e. such that g = r ⊕ h asvector spaces (but in general not as Lie algebras!).

Example 11.8. For g = gln(K) the subalgebra h := sln(K) is a Levi subal-gebra. Note that in this case

gln(K) = r⊕ sln(K)

even as Lie algebras.

Remark 11.9. Given a Levi-Malcev-decomposition g = r ⊕ h of the Liealgebra g = Lie(G) of the simply connected Lie group G, the group G isthe semidirect product of the simply connected Lie groups R,H of r and h.Indeed

G ∼= R×σ Hwith the group homomorphism σ : H −→ Aut(r) ∼= Aut(R), h 7→ Ad(h).

But note that GLn(K) in general is not isomorphic to K∗E × SLn(K),instead one has to factor out the finite subgroup

K∗E ∩ SLn(K) ∼= Cn(K)

with the group of n-th roots of unity Cn(K) := a ∈ K; an = 1. So

GLn(K) ∼= (K∗E × SLn(K))/Cn(K)E.

For the further study of Lie algebras it turns out to be useful to investigatein general homomorphisms g −→ gl(V ) for a K-vector space V (also calledrepresentations of g in V ). Slightly changing the point of view (but not theobjects under consideration!) we arrive at the notion of a g-module:

Definition 11.10. Let g be a Lie algebra.

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1. A (finite dimensional) g-module V is a (finite dimensional) K-vectorspace V together with a bilinear map

g× V −→ V, (X, v) 7→ Xv,

such that [X, Y ]v = X(Y v)− Y (Xv).

2. A g-module is called

(a) irreducible or simple if V and 0 are the only g-submodules.

(b) semisimple if any g-submodule U ⊂ V admits a complementaryg-submodule W ⊂ V , i.e. such that V = U ⊕W .

3. A Lie algebra g is called semisimple, if all its finite dimensional g-modules are semisimple.

So g-modules are in one-to-one correspondence with K-vector spaces Vtogether with a Lie algebra homomorphism g −→ gl(V ). In particular, re-garding K as abelian Lie algebra, a K-module is not just a K-vector space,but rather a pair (V, f) with a K-vector space V and an endomorphismf ∈ gl(V ): The Lie algebra homomorphisms K −→ gl(V ) are uniquelydetermined by their value f ∈ GL(V ) at 1 ∈ K, and that value can beprescribed arbitrarily.

For solvable Lie algebras we have:

Theorem 11.11 (Theorem of Lie). Any (finite dimensional) module V overa solvable complex Lie algebra g admits a one dimensional submodule L ⊂ V .In particular an irreducible g-module over a solvable complex Lie algebra ghas dimension dimV = 1.

Corollary 11.12. 1. Any (finite dimensional) module V over a solvablecomplex Lie algebra g admits an invariant flag

0 = V0 ⊂ V1 ⊂ ... ⊂ Vn = V

of submodules Vi ⊂ V of dimension i.

2. A solvable complex Lie algebra is isomorphic to a subalgebra of utn(C)for some n ∈ N.

Before we discuss semisimple algebras and modules let us study an im-portant subclass of all solvable algebras:

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Definition 11.13. A Lie algebra g is called nilpotent if the following de-creasing sequence of subalgebras N i(g) terminates at 0:

N0(g) := g, N i+1(g) = [g, N i(g)].

Remark 11.14. We have N(g) = C(g) and Ci(g) ⊂ N i(g) for i > 1, hencea nilpotent algebra is solvable.

Example 11.15. 1. The Lie algebra uun(K) consisting of all upper tri-angular matrices with zeros on the diagonal is nilpotent.

2. The Lie algebra g = KX + KZ with [X,Z] = Z is solvable, but notnilpotent: We have N i(g) = KZ for all i ∈ N>0.

3. Abelian Lie algebras are nilpotent.

4. Subalgebras and factor algebras of nilpotent Lie algebras are nilpotent.

In a nilpotent Lie algebra g the Baker-Campbell-Hausdorff series

C(X, Y ) =∞∑i=1

Ci(X, Y ) = (X + Y ) +1

2[X, Y ] + C3(X, Y ) + ...

is finite and hence defines a polynomial map

C : g× g −→ g.

Indeed, it provides g with a group law: Consider the exponential map exp :g −→ G for the simply connected Lie group with Lie(G) ∼= g. Then, expbeing diffeomorphic near 0 ∈ g, the conditions for a group law are satisfiednear the origin and hence everywhere by the identity theorem for polynomialmaps (A map V ×V −→ K for a K-vector space V is called polynomial if it isa K-linear combination of products of linear forms in one of both variables. Amap V ×V −→ V is polynomial, if the composition with any linear functionalV −→ K is polynomial.). Indeed the exponential map exp : g −→ G turnsout to be an isomorphism of Lie groups. So we can replace G with (g, C(., .)),the expression for C in terms of Lie monomials being independent from thenilpotent Lie algebra g. Note that the n-th power of X ∈ g for n ∈ Z withrespect to C(., .) is nX.

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Example 11.16. The exponential map exp : uun(K) −→ UUn(K) is poly-nomial:

exp(X) =n−1∑i=0

X i

i!

and has the inverse

log : UUn(K) −→ uun(K), log(Y ) =n−1∑i=1

(−1)i+1 (Y − E)i

i.

In order to understand all Lie groups with nilpotent Lie algebra we haveto consider factor groups of (g, C(., .)) mod central discrete subgroups D ⊂(g, C(., .)). We claim that the center of the Lie group (g, C(., .)) is the centerker(ad) of the Lie algebra g: Since C(X, 0) = 0 = C(0, Y ), any of theLie bracket monomials in C(X, Y ) contains both X and Y as factor, henceC(Z,X) = Z + X = X + Z = C(X,Z) for Z ∈ ker(ad) and any X ∈ g.On the other hand, for a central element Z of the Lie group (g, C(., .)) all itsintegral powers nZ are central as well; hence C(nZ,mX) = C(mX.nZ) forall n,m ∈ Z. Both expressions being polynomials in n,m ∈ Z, comparisonof the bilinear term yields [Z,X] = [X,Z] resp. [Z,X] = 0. So centraldiscrete subgroups are exactly the lattices in the subspace ker(ad) ⊂ g. Notefurthermore that the connected Lie subgroups of (g, C(., .)) are exactly thesubalgebras h ⊂ g (the exponential map being the identity on g and subspacesbeing maximal connected submanifolds).

Theorem 11.17. 1. A Lie algebra g is solvable iff its commutator algebraC(g) is nilpotent.

2. A Lie algebra is nilpotent iff ad(X) ∈ gl(g) is nilpotent for every X ∈ g.

3. Any nilpotent Lie algebra is isomorphic to a subalgebra of uun(K) forsome n ∈ N.

Example 11.18. A non-linear Lie group: Consider G = (g, C(., .))/D with

D = Z · [X, Y ] with a central element [X, Y ]. Assume ϕ : G/D∼=−→ H ⊂

GLn(K) is an isomorphism of Lie groups. Then, g being solvable we mayassume h ⊂ utn(K), see Theorem 11.11 resp. H ⊂ UTn(K), see Cor. 11.12.Consider the element Z := 2−1[X, Y ]. Since C(utn(K)) ⊂ uun(K) we findϕ∗(Z) ∈ uun(K), hence ϕ(Z) = exp(ϕ∗(Z)) ∈ UUn(K) (note that G is here

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identified with its Lie algebra), but UUn(K) contains no non-trivial elementsof finite order!

Now let us consider semisimple algebras and modules:

Corollary 11.19. A semisimple g-module V is the direct sum of irreducibleg-modules

V =s⊕i=1

Vi,

the factors V1, ..., Vs being unique up to isomorphy and order.

But note that for v ∈ V the subspace gv := Xv;X ∈ g in general is nota (g-)submodule, so the irreducible factors are not necessarily factor algebrasof g (e.g. X(Y v) need not belong to gv). On the other hand V := g is ag-module with the Lie bracket as ”scalar multiplication” (corresponding tothe adjoint representation g −→ gl(g), X 7→ ad(X)). Then the irreduciblefactors are ideals of g. Calling a Lie algebra g simple if it is semisimple andadmits no nontrivial ideals we obtain:

Theorem 11.20. A semisimple Lie algebra g is the direct sum of simple Liealgebras

g =s⊕i=1

gi,

the factors g1, ..., gs being unique up to isomorphy and order.

Ideals of semisimple algebras are direct factors and thus semisimple aswell. As a consequence, no nontrivial solvable algebra is semisimple, sinceotherwise we would find that the one-dimensional Lie algebra K is semisim-ple. Furthermore, a semisimple algebra has trivial radical. Indeed the reverseimplication holds as well:

Theorem 11.21 (Theorem of Weyl). A Lie algebra g with trivial radical issemisimple.

Example 11.22. 1. A real Lie algebra g, such that the simply connectedLie group G with Lie(G) ∼= g is compact, is semisimple: The Lie algebrahomomorphism g −→ gl(V ) defining a g-module is induced from aLie group homomorphism ϕ : G −→ GL(V ). According to the nextpoint there is an inner product σ : V × V −→ K, such that all maps

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ϕ(g) : V −→ V, v 7→ gv := ϕ(g)v are σ-isometries. Now, given a g-submodule U ⊂ V , the σ-orthogonal complement W := U⊥ is bothG-invariant and a complementary g-module. So son(R) and sun aresemisimple for n > 1.

We remark, that any real semisimple algebra is of the type describedabove.

2. Invariant inner products: On any Lie group G there is a left-invariantσ-finite measure µ on the Borel subsets of G, the ”Haar measure”,unique up to a scalar factor. For compact G, the Haar measure is evenright invariant and µ(G) < ∞. Then we may take any inner productτ : V × V −→ K in order to get by means of averaging over G thedesired G-invariant inner product:

σ(v, w) :=

∫G

τ(gv, gw)dµ(g).

3. Complex semisimple Lie algebras: The first point applies only to realLie algebras, since there are no compact simply connected Lie groupsexcept the trivial group. (The only connected compact complex Liegroups are the tori G = Cm/Λ with a lattice Λ ∼= Z2m of maximal rank.)But we can weaken our assumption: It is sufficient that g = k⊕ ik witha real Lie subalgebra k ⊂ g belonging to a simply connected compactreal Lie group K (not to be confused with the base field). Given now ag-submodule U ⊂ V , its orthogonal complement with respect to a K-invariant inner product on V is a K-invariant complex vector subspace,hence also k and g = k + ik-invariant. As example take

g = sln(C) = sun ⊕ isun.

org = son(C) = son(R)⊕ ison(R).

Again, any complex semisimple Lie algebra is obtained in that way.

4. Haar measure (Only for those knowing differential forms): Let G bea real Lie group of dimension n. As with vector fields, any n-formωe ∈

∧n T ∗eG extends uniquely to a left invariant n-form ω ∈ Ωn(G).Then µ(A) :=

∫Aω (using the orientation of G defined by ω itself)

provides a Haar measure on the Borel subsets A ⊂ G.

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In the remaining part of this section we explain the classification of com-plex simple Lie algebras.

Definition 11.23. A subalgebra h ⊂ g of a Lie algebra g is called a Cartansubalgebra (CSA), if it is nilpotent and satisfies

[X, h] ⊂ h =⇒ X ∈ h, ∀ X ∈ g.

Example 11.24. For g = sln(C) the subalgebra h := sdn(C) consisting ofall diagonal matrices in sln(C) is a CSA.

Proposition 11.25. In a complex Lie algebra g any two CSA h, h′ ⊂ g areconjugate under an automorphism f = ead(X) for some X ∈ g, i.e. h′ = f(h).

For a semisimple algebra we have

Proposition 11.26. Let g be a complex semisimple algebra. Then a subal-gebra h ⊂ g is a CSA if and only if the following conditions are satisfied:

1. The subalgebra h ⊂ g is a maximal abelian subalgebra.

2. All homomorphisms ad(H) : g −→ g are diagonalizable.

Since h is abelian, the endomorphisms ad(H) for H ∈ h commute one withthe other, hence can be diagonalized simultaneously, and satisfy ad(H)|h = 0.So defining gα ⊂ g for a linear form α : h −→ C by

gα := X ∈ g; ad(H)(X) = α(H)X, ∀ H ∈ h

we can write

g = h⊕⊕α∈∆

gα

with the following finite subset ∆ ⊂ h∗:

∆ := α ∈ h∗ \ 0; gα 6= 0.

The case α = 0 does not occur, h being a maximal abelian subalgebra.

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Example 11.27. For g = sln(C), h = sdn(C)(:= the diagonal matrices withtrace 0) we have

∆ = αij := βi − βj; 1 ≤ i, j ≤ n, i 6= j

whereβi : sdn(C) −→ C, D = (zkδk`) 7→ zi

withgij := gαij

= CEijwith the matrix Eij := (εk` := δkiδ`j).

Since ad(H) ∈ Der(g) we find

[gα, gβ] ⊂ gα+β,

indeed[gα, gβ] = gα+β.

Furthermore one can show

dim gα = 1, ∀ α ∈ ∆

and that∆ ∩ Cα = ±α, ∀ α ∈ ∆.

The set ∆ spans a real subspace

h∗R :=∑α∈∆

Rα ⊂ h∗

with h∗ = h∗R ⊕ ih∗R. For a more detailed description of ∆ we need a naturalinner product on h∗R.

First of all on a Lie algebra g one defines:

Definition 11.28. Let g be a Lie algebra. The Cartan-Killing form is thefollowing bilinear symmetric form

〈., .〉 : g× g −→ K, (X, Y ) 7→ 〈X, Y 〉 := Tr(ad(X)ad(Y )).

Let us mention:

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Theorem 11.29. Let g be a Lie algebra with Cartan-Killing form 〈., .〉 :g× g −→ K. Then g is

1. solvable iff 〈g, C(g)〉 = 0 and

2. semisimple iff its Cartan-Killing form is nondegenerate.

Moreover in the latter case its restriction to h × h with a CSA h ⊂ g isnondegenerate as well, and its dual form, also denoted

〈., .〉 : h∗ × h∗ −→ C,

is real valued on h∗R × h∗R and even positive definite.

Remark 11.30. Define Hα ∈ h by 〈Hα, H〉 = α(H). Then

[X, Y ] = 〈X, Y 〉Hα 6= 0

for X ∈ gα \ 0, Y ∈ g−α \ 0.

In particular we see that

CHα ⊕ gα ⊕ g−α ∼= sl2(C).

The classification of complex semisimple Lie algebras now depends on abetter understanding of the set ∆ ⊂ h∗R. Indeed it has a remarkable symmetryproperty: It forms a root system:

Definition 11.31. A finite subset ∆ ⊂ E \ 0 of a finite dimensional eu-clidean vector space E with inner product 〈., .〉 is called a root system if thefollowing conditions are satisfied:

1. E = span(∆)

2. ∆ ∩ Rα = ±α for all α ∈ ∆.

3. For any root (i.e. element) α ∈ ∆ the reflection

sα : E −→ E, v 7→ v − 2〈v, α〉〈α, α〉

α

on the hyperplane α⊥ ⊂ E leaves ∆ invariant:

sα(∆) = ∆, ∀ α ∈ ∆.

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4. For all β, α ∈ ∆ we have

χ(β, α) :=2〈β, α〉〈α, α〉

∈ Z.

The fourth condition is a very strong condition on the possible anglesbetween the roots: Denote ϑ ∈ [0, π) the angle between the non proportionalroots α, β. Then

χ(β, α)χ(α, β) = 4 cos2(ϑ) ∈ Z,

hence cos(ϑ) ∈ 0,±12,± 1√

2,±√

32 resp. ϑ = 0, π

6, π

4, π

3, π

2, 2π

3, 3π

4, 5π

6.

Furthermore χ(β, α) = ±1,±2,±3, if α, β are neither proportional nororthogonal. A more detailed analysis of that situation shows that in case||β|| ≥ ||α|| we have

||β||2 = 4 cos2(ϑ) · ||α||2 = χ(β, α)χ(α, β)||α||2.

Indeed, that means nothing but β = ±(α − sβ(α)), the sign depending onwhether ϑ < π

2or ϑ > π

2.

The proof that the above conditions hold for ∆ ⊂ h∗ relies on the knowl-edge of the irreducible sl2(C)-modules. Indeed, given nonproportional α, β,the subspace ⊕

k∈Z

gβ+kα

is a module over the subalgebra

CHα ⊕ gα ⊕ g−α ∼= sl2(C).

Now let us concentrate on abstract root systems:

Definition 11.32. Let ∆ ⊂ E be a root system, and demote O(E) the groupof linear isometries of the euclidean space E. The Weyl group W (∆) ⊂ O(E)is defined as the subgroup of O(E) generated by the reflections sα, α ∈ ∆.

The symmetries of a root system ∆ given by the action of the Weyl groupW (∆) make it possible to compress the information contained in it in a basisB:

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Definition 11.33. A subset B ⊂ ∆ is called a basis of the root system∆ ⊂ E if B is a basis of the vector space E and every β ∈ ∆ is an integrallinear combination

β =∑α∈B

kα · α,

where the coefficients kα ∈ Z satisfy either kα ≥ 0 for all α ∈ B or kα ≤ 0for all α ∈ B.

A basis of a root system ∆ ⊂ E gives rise to a decomposition

∆ = ∆B + (−∆B),

with ∆B := ∆ ∩ (∑

α∈B N≥0α). On the other hand, starting with certaindecompositions we get all the bases of a root system ∆:

Proposition 11.34. Let ∆ ⊂ E be a root system and H ⊂ E a hyperplanewith H∩∆ = ∅. Given a connected component E0 of E\H the indecomposableelements in ∆ ∩ E0, i.e. those which can not be written as a sum β1 + β2

with β1, β2 ∈ ∆ ∩ E0, constitute a basis of the root system ∆. Indeed, anybasis of ∆ is obtained in that way. In particular any root can be realized asan element of a suitable basis B ⊂ ∆. Furthermore the angle between twobase vectors is obtuse, i.e. ∈ [π

2, π).

Proposition 11.35. Let ∆ ⊂ E be a root system.

1. The action of the Weyl group W (∆) on the set of bases of ∆ is simplytransitive.

2. Given a basis B, the reflections σα, α ∈ B, generate W (∆).

As a consequence we see that |W (∆)| < ∞. Furthermore that given abasis B we can recover the Weyl group W (∆) as well as ∆ = W (∆)B.

Let us explain a little bit more in detail the different bases a root system∆ admits: Writing a hyperplane as H = Pγ := γ⊥ for γ ∈ E we see that itis a separating hyperplane for ∆, i.e., Pγ ∩∆ = ∅, iff γ 6∈ E \

⋃α∈∆ Pα.

Definition 11.36. Let ∆ ⊂ E be a root system. The connected componentsof E\

⋃α∈∆ Pα are called Weyl chambers. An element γ ∈ E is called regular

if γ 6∈ E \⋃α∈∆ Pα. For such an element γ denote Ch(γ) the Weyl chamber

containing γ.

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Proposition 11.37. For a regular element γ denote Bγ ⊂ ∆ the unique basisof ∆ with 〈γ,Bγ〉 > 0. Then Bδ = Bγ for all δ ∈ Ch(γ) and Ch(γ) 7→ Bγ isa bijection between the set of Weyl chambers of ∆ and the set of bases of ∆.

Now the information contained in a basis B ⊂ ∆ can be encoded in aso called Dynkin diagram, a graph whose vertices are the base roots α ∈ B.Two vertices α, β are connected by χ(β, α)χ(α, β) = 4 cos2(ϑ) edges, i.e. byone edge, if the angle ϑ ∈ [0, π) equals 2π

3, by two edges, if ϑ = 3π

4and by

three edges if ϑ = 5π6

. In the last two cases the two or three edges are evenoriented, the arrow pointing from the longer root to the smaller one. Notethat the diagram does not depend on the choice of the basis B.

Let us come back to Lie algebras: A Lie algebra can be reconstructed -up to isomorphy - from its root system, and a root system from one of itsbases resp. - again up to isomorphy - from its Dynkin diagram. First of all,it is connected if and only if the corresponding algebra is simple, and thereis a complete classification of the connected Dynkin diagrams. Here it is, theindex counting the number of vertices:

1. A`, ` ≥ 1: A linear string with ` vertices and only simple edges.

2. B`, ` ≥ 2: A linear string with ` vertices with ` − 2 simple edges andone double edge at one of its ends, the arrow pointing to the end point(for ` ≥ 3).

3. C`, ` ≥ 3: A linear string with ` vertices with `−2 simple edges and onedouble edge at one of its ends, the arrow pointing to the inner point.Note that C2 = B2.

4. D`, ` ≥ 4: A string with `− 2 vertices, with the two remaining verticesconnected to the same end point of the string. All edges are simple.Note that D3 = A3.

5. E`, ` = 6, 7, 8: A string with ` − 1 vertices, the `-th vertex being con-nected to the third vertex of the string. All edges are simple.

6. F4: A linear string with 4 vertices, the outer edges being simple, theinner one being a double edge (the orientation is not important, sinceboth choices give isomorphic Dynkin diagrams).

7. G2: Two vertices connected by three edges.

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Indeed, for every of the above Dynkin diagrams, there is a complex simpleLie algebra realizing it.

Now let us look at complex semisimple Lie groups, i.e. complex connectedLie groups G with semisimple Lie algebra g = Lie(G).

First of all the adjoint representation Ad : G −→ GL(g) is a covering ofthe subgroup Ad(G) ⊂ Aut(g) ⊂ GL(g), since its differential ad : g −→ gl(g)is injective for a semisimple algebra g, its center being trivial. So there is notonly a maximal Lie group - the simply connected one - but also a minimalLie group with Lie algebra g, since Ad(G) only depends on g: It is theconnected Lie subgroup of Aut(g) with Lie algebra ad(g). Indeed, for asemisimple algebra we have ad(g) = Der(g) and hence Ad(G) = Aut0(g), thecomponent of the identity of Aut(g).

If G is simply connected, any other connected Lie group with Lie algebrag is of the form G/D with a discrete subgroup D ⊂ Z(G). Indeed thecenter Z(G) ⊂ G is finite, it can even be read off from the root system∆ associated to a CSA h ⊂ g = Lie(G): If we denote Γ0 ⊂ E := h∗R thelattice generated by ∆ (in fact Γ0 =

⊕α∈B Zα with any basis B ⊂ ∆) and

Γ := γ ∈ E;χ(γ,∆) ⊂ Z, then, obviously Γ0 ⊂ Γ and (less obviously)

Z(G) ∼= Γ/Γ0.

The connected Lie subgroup H ⊂ G is a maximal (complexified) torus (C∗)`,a closed subgroup of G containing the center Z(G).

Furthermore semisimple Lie groups can be realized as algebraic (in par-ticular closed) subgroups of GL(V ) for some complex vector space V ; andeven better, any homomorphism G −→ GL(W ) for an arbitrary vector spaceW is algebraic. Indeed this follows from the knowledge of all irreducibleg-modules W resp. all irreducible representations g −→ gl(W ).

Here is the table of all simply connected complex simple Lie groups:

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Dynkin diagram simply connected Lie group Dimension Center

A`, ` ≥ 1 SL`+1(C) `(`+ 2) Z`+1

B`, ` ≥ 2 Spin2`+1(C) `(2`+ 1) Z2

C`, ` ≥ 3 Sp2`(C) `(2`+ 1) Z2

D`, ` ≥ 4, even Spin2`(C) `(2`− 1) Z2 × Z2

D`, ` ≥ 5, odd Spin2`(C) `(2`− 1) Z4

E6 — 78 Z3

E7 — 133 Z2

E8 — 248 0F4 — 52 0G2 Aut(O(C)) 14 0

Here Spinn(C) denotes the universal covering group of SOn(C), and O(C)is the complexified algebra of Cayley numbers (octonians). The remainingexceptional group do not have an immediate geometric realization.

Note that an arbitrary semisimple complex Lie group is of the form G/D,where G is a finite product of copies of the above Lie groups and D ⊂ Z(G),with Z(G) being the direct product of the centers of the simple factors.

Complex semisimple and real compact Lie groups: Finally letus comment on the relation between complex semisimple groups and realcompact Lie groups. Let us start with a semisimple Lie algebra g and lookfor a ”real compact form” k ⊂ g, i.e., a real subalgebra k ∼= Lie(K) for somereal compact Lie group K satisfying g = k⊕ ik. We start with a CSA h ⊂ gand write

g = h⊕⊕α∈∆

gα,

noting that, with respect to the Cartan Killing form 〈., .〉 we have h ⊥ gα forall α ∈ ∆ as well as gα ⊥ gβ for α+ β 6= 0. We shall represent k = Fix(τ) asfix algebra of a conjugation

τ : g −→ g,

i.e. an involutive automorphism of g as real Lie algebra (τ 2 = idg) satisfyingτ(iX) = −iX. Setting

g0 = hR ⊕⊕α∈∆

RZα ⊂ g

with hR := span(Hα;α ∈ ∆) (see Remark 11.30) we have g = g0 ⊕ ig0 and aconjugation σ := idg0 ⊕−idig0 .

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We want to take τ = ϕ σ with a complex automorphism ϕ : g −→ g: Inorder to define ϕ, we choose elements Zα ∈ gα, Z−α ∈ g−α with 〈Zα, Z−α〉 =−1. Then the linear map ϕ : g −→ g with ϕ|h = −idh and ϕ(Zα) = Z−α isthe desired Lie algebra automorphism.

Example 11.38. For g = sln(C), h = sdn(C) we have σ(A) = A and ϕ(A) =−AT .

Since X = iH +∑cαZα ∈ k satisfies H ∈ hR and cα = c−α we find

〈X,X〉 = −〈H,H〉 − 2∑α∈∆

|cα|2 < 0,

so the negative Cartan-Killing form −〈., .〉 is positive definite on k. So wehave obtained an inner product on k invariant under any automorphism of k(an automorphism of k extends to an automorphism of g). Hence the closedsubgroup Aut(k) ⊂ O(k,−〈., .〉) is compact. On the other hand one can provethat Lie(Aut(k)) = Der(k) = ad(k), whence k = Lie(K) with K := Aut(k).

Theorem 11.39. Given a complex semisimple algebraic group G there isa compact real Lie subgroup K ⊂ G (unique up to conjugacy) such thatk := Lie(K) is a compact real form of g := Lie(G), and the map

K × ik −→ G, (x,X) 7→ x exp(X)

is a diffeomorphism.

Trevlig Sommar!

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